Tom Rexton

cardinality of infinite sets

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Is the concept of cardinality of infinite sets valid? In my math course our last unit covers cardinality, and we prove basic theorems about set cardinalities. I'm having trouble determining whether it is valid to apply the concept of "equinumerosity" to infinite sets; that is, set A is "equinumerous" with set B if there is a bijective function from A to B. This definition is used to prove such theorems as R is "more infinite" than N, Z or Q, or formally |R|>|Z|=aleph-null. (All the proofs we've done rest on the validity of constructing a bijection from Z+ to some other infinite set, and showing whether it is possible or impossible to do so, and thereby prove that it is equinumerous to or "more infinite" than Z+.)

Perhaps there's a valid way of thinking about infinite cardinalities of different sizes? Or does it depend on the invalid assumption of an actual infinity?

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Is the concept of cardinality of infinite sets valid? In my math course our last unit covers cardinality, and we prove basic theorems about set cardinalities.  I'm having trouble determining whether it is valid to apply the concept of "equinumerosity" to infinite sets; that is, set A is "equinumerous" with set B if there is a bijective function from A to B.  This definition is used to prove such theorems as R is "more infinite" than N, Z or Q, or formally |R|>|Z|=aleph-null.

I doubt that you intend to question the validity of bijective functions, so what do you mean by "valid" in regard to the cardinality of infinite sets?

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I doubt that you intend to question the validity of bijective functions, so what do you mean by "valid" in regard to the cardinality of infinite sets?

Can we define the cardinality of infinite sets in that manner? I can see why it's valid for finite sets, but not for infinite sets. How can it make sense to say that one infinite set is more infinite than another? In other words, how does it make sense to talk of degrees of infinity? (e.g., aleph-null, 2^(aleph-null), etc.)

It's deducible that there cannot be a bijection from Z+ to R. Yet is it valid to concluded that |R| >|Z+| when they're both infinite? Since "countable" is defined simply as either finite or denumerable, and since R is neither, we can conclude that R is uncountable. But in what way does this fact make it have more elements than a set that is already infinite? I would think we can conclude at most that R is uncountable.

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I doubt that you intend to question the validity of bijective functions, so what do you mean by "valid" in regard to the cardinality of infinite sets?

Can we define the cardinality of infinite sets in that manner? I can see why it's valid for finite sets, but not for infinite sets. How can it make sense to say that one infinite set is more infinite than another? In other words, how does it make sense to talk of degrees of infinity? (e.g., aleph-null, 2^(aleph-null), etc.)

I'm sure that when ewv gets around to seeing the thread, he will chime in. He has some very definite ideas on this. Let me just say that I think you are right to question what facts of reality underlie such a one-to-one correspondence between infinite sets. Others disagree, but I personally think that no direct connection can be made between that process and concrete reality. However, such a direct connection is not a necessary requirement for a concept of method, and the logical basis, though not approached that way by Cantor, is neither arbitrary nor useless. Aspects of measure theory and topology alone depend these results.

If, on the other hand, you are thinking in terms of foundational mathematics, then your question of "sense" cannot be offset by practical uses in higher-level math. Cantor was both a loony and a genius. He pledged his allegiance to the worst of Leibniz and he thought that actual monads were the irreducible elements of the world. His aleph series were transcendental changes leading to a higher power. Cantor was one who saw the actual infinite in God, represented as transfinite in the contingent world of creatures and in thought via mathematical magnitude. Nevertheless, he built an edifice that has served a purpose, even with a foundation as shaky as it is.

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Sorry for the long delay, but this subject takes more than a casual response and I have been infinitely busy (not more than aleph_0, but it seems that way). Stephen is correct that I have definite views on this subject and I hope to clarify them and answer Tom's questions, but I agree with Stephen's general statement, "no direct connection can be made between that process and concrete reality. However, such a direct connection is not a necessary requirement for a concept of method, and the logical basis, though not approached that way by Cantor, is neither arbitrary nor useless."

The last time I tried to explain this subject the explanation came out too terse for at least some readers and I was accused of "gobbledygook" :-(, so this one will be longer (but despite its initial appearance, still finite). Even with that I will omit technical details, referring informally to results commonly explained in standard texts since the point of this is to answer Tom's philosophical questions, not to provide a complete exposition on the subject, and he (and others) have access to the technical references with basic results. So this will still require a certain background knowledge and motivation to make the connections (and anyone reading this who is not familiar with the technical background may still understandably think it sounds like gobbledygook), but this time I will try to spell out the explanations and implications of the basic ideas more explicitly, including the basic role of 1-1 mappings that Stephen once wondered about for this application.

Tom, you are right to question the concepts and theory of infinite cardinals because infinite sets have no numerical "size". Because they are infinite they are larger than any number, so if you think of "size" as an ordinary number there are no sizes of infinite sets to compare or to order. But the concept of cardinality of infinite sets is based on a different, more abstract idea than a simple number, involving the complexity of combinatorics applied to infinite sets.

1. Any time you deal with the infinite you have to mean the "mathematical infinite", in the sense discussed in this forum several times, or it makes no sense at all. You cannot ever treat infinity as an actual number, or an infinite set as an actual completed whole, and that goes for Cantor's cardinals, too.

Cantor believed in the actual infinite and thought his infinite cardinals were real, including -- beyond all the other more mundane infinities -- a super infinity that I think he once equated with God. But you already know that such Platonism has to be wrong. The important issue to discuss here is the meaning of mathematical principles.

Infinity in mathematics is always a concept of method, a high level abstraction depending on some particular structure of how something grows or subdivides indefinitely, but through some finite process. Using infinity in mathematics requires a solid understanding of that principle (for example, how the natural numbers are incremented in steps of one, how an exponential function grows, how successive terms in an infinite series are related, how an interval is subdivided in analyzing continuity or derivatives, or what happens when you reverse the order of two limits.) Much of mathematical analysis is concerned with whether some infinite sequence or class of sequences converges or becomes unbounded (or simply wanders without converging). A finite mechanism behind some infinite sequence -- how the process works -- is the only way you can analyze, compare or combine infinite sets. The process of "how", in terms of finite comparisons, is the essence of it, without which the infinite becomes a floating abstraction.

2. Now turn to the concept of enumeration. If you try to compare the "natural numbers" (1, 2, 3, ...) with the "even numbers" (2, 4, 6, ...), you might think that there are only half as many even numbers because you skip every other number. That is true for a finite set, but when you indefinitely count the even numbers in an open-ended process by indexing them with positive integers, the process continues without end: There are no numerical sizes of the infinite sets for one to be twice as big as the other.

All the integers (positive and negative) can also be indexed by the positive integers as a list in the order 0, 1, -1, 2, -2, ... Again, there are not "half as many" positive numbers just because you skip over the negative numbers. There is no "size" to be half of. As another example, Galileo saw that the sequence of the squares of positive integers has a 1-1 correspondence with the positive integers themselves (1 <-> 1=1^2, 2 <-> 4=2^2, 3 <-> 9=3^2, ...), even though there are "fewer" perfect squares -- known as Galileo's paradox.

But it isn't a paradox: you can index by counting elements in any finite or infinite set, using positive integers to designate their place in a list, including subsets of the integers themselves.

The general concept of integer indexing -- of a one-to-one mapping from the positive integers to other mathematical entities as a method for keeping track of them -- is used throughout mathematics and its applications. You see it all the time for terms in infinite sequences and series involving variables, functions, coordinates systems or transforms between coordinates, etc. For example, infinite sets of solutions of differential equations (such as eigenvalues and eigenfunctions) are routinely indexed to keep track of them.

The abstract use of integers as indices is a very common method, amounting to a form of measurement of the abstract entities in either finite or open-ended sets. The fact that physical measurements of objects referred to directly or indirectly by mathematical terms don't come with metaphysical indices physically attached is not relevant. Mathematics is a science of method, not things.

3. Having seen some examples of indexing different kinds of integers like even numbers and squares, you might think that this cannot be done with the rational numbers (fractions) because there are "too many" of them -- not just twice as many but infinitely many more. They are infinitely subdividing -- by the increasing denominators -- while simultaneously infinitely growing -- as the numerators increase. But as Tom has seen in his course, they too can be indexed by a special method: tracing a progression of diagonals (NE to SW) across a 2x2 array in which the rows are indexed by the numerators and the columns are indexed by the denominators. As the index goes back and forth across the diagonal "front" that sweeps down and across the array to the SE, it systematically covers all the fractions (including the improper fractions bigger than 1).

There is more than one way to do the indexing, and the order in the resulting lists does not correspond to the numerical order of the size of the fractions, but any of these schemes show that the rational numbers can be systematically enumerated so that there are a finite number of steps counting from the start to any rational number.

The same approach can be used for any ordered pairs (nx,ny), not just fractions in the form (numerator,denominator): You can enumerate any Cartesian cross product of any two enumerable sets. The coordinates of integer lattice points in 3-D space can then be enumerated as triplets (nx,ny,nz), first by establishing an enumeration of the nx and ny coordinates and then enumerating the pairs (nx,ny) in one list, paired with nz in the second list. Any triplets of enumerated sets can be enumerated, and that implies, inductively, that any finite n-tuple consisting of elements from enumerable sets can be enumerated by repeating the same process used for the fractions.

4. The question then arises if there is anything that cannot be enumerated. Cantor showed, first by a more complex argument of mathematical analysis, and later with his famous "Cantor diagonal argument", that the real numbers cannot be enumerated: there is no way to go about systematically listing them so that they are all accounted for, with a finite number of steps indexing any real number. The diagonal argument shows that when you consider the real numbers, even restricted to the interval [0,1], represented in decimal form .d1 d2 d3 ... , you can prove that the hypothesis that they can be listed systematically, indexed by the integers, leads to a contradiction, so the hypothesis must be false. The contradiction is demonstrated by showing how to specify a sequence of digits .d1 d2 d3 ... that cannot be part of the list.

No method of enumeration like the one used for rational numbers or anything else will work for the real numbers.

You can "index" the real numbers just by picking them one at a time in any order you choose, adding them to a list as you go along, but there is no way to systematically index them that in principle guarantees that none will be skipped, i.e., that guarantees in principle that any number selected is already accounted for by the systematic method, so that given any real number you can guarantee in advance that it can reached in a finite number of steps using the enumeration scheme.

Since the decimal representations of real numbers are themselves infinite strings of digits, this also means, by the same diagonal argument, that while you can enumerate any n-tuples for any specific number n, you can't necessarily do it when n itself becomes infinite with its own limit process: any (n1,n2,n3,...,N) works but (n1,n2,n3,...) does not. Note the double limit, one for the dimension of the n-tuple and the other for the list.

The complexity of the combinations of digits in decimal real numbers is too much for an integer indexing scheme to systematically keep track of them; any such method would necessarily be overwhelmed -- there are too many digits accumulating in too many different ways to "keep up" with. Enumerating all the infinite decimals would require an infinity of elements, each of which contains an infinity of digits in the sequence of decimal places. The nature of the double progression is too complex for simple enumeration to deal with.

But that is not true for all infinite decimals. Rational numbers can be enumerated in terms of their numerators and denominators, and the fractions are equivalent to their own infinite decimal expansions. But when you express the rational numbers in decimal form there are always infinitely repeating finite patterns, like the sequences .333333... or .5000000 or .5671212121212... No matter how complex the pattern, it is always finite. The repetition of finite patterns limit the possibilities of the combinations of digits, and that is enough to allow the rationals to be enumerated. But for irrational numbers, any combination of unending digits is possible, which is too much, in principle, for any systematic integer indexing scheme.

Since the higher level abstraction of irrational numbers is based on representing irrational numbers by sequences of rational numbers of indefinitely increasing precision, expression of irrational numbers in decimal form require the set of all sets of non-negative integers (the digits): You need all the possible combinations of numbers in all possible sequences. Non-denumerability of the real numbers means there is no bijection between the set of non-negative integers and its power set, the set of all its subsets.

The number of subsets of elements taken in all combinations from a finite set of n elements is 2^n. That is why the cardinal for the reals is denoted with the "power notation" aleph_1 = 2^aleph_0. The fact that the real numbers are not enumerable is due to the cumulative complexity of the combinatorics inherent in the abstraction of the continuum as expressed by infinite decimal expansions on the number line.

But aleph_0, the infinite cardinality of the natural numbers, is not a number (and neither is any other cardinal). The symbol 2^aleph is only a means of notation for an abstraction of method based on the complexity of the combinatorics; it is not literally 2 to some number as the exponent.

(If a whole number is "a mental symbol that integrates units into a single larger unit" [AR in IOE], then an infinite cardinal cannot be a number because it refers, epistemologically, to an unlimited number of units with respect to some method of progression; The infinity of units -- larger than any number -- that you would have to integrate into one unit as a cardinal number does not exist as a completed whole. You can form an abstract concept of an infinite process, but whatever you mentally integrate to do that, it does not fall under the concept of number. However infinity is used in mathematics, it cannot be a number, despite Cantor's "arithmetic" of infinities.)

To summarize so far, even though you can try to index the real numbers just by picking them one at a time at random for as far as you want to go, with the potential to always go farther, the technical theory of non-denumerability is concerned with systematic schemes that in principle guarantee in advance that nothing is missed, and the reasoning behind it is the complexity of the "2^n" combinatorics as the sequences progress indefinitely and the impossibility of a bijection between a set and its power set (even though Cantor's argument was indirect as a reductio ad absurdum).

Those subtleties may seem more complex than Cantor's simple Platonic view of actual infinite cardinals bouncing around counting infinite sets as completed wholes, but that is the price you pay to understand the mathematics free of mysticism or subjective floating abstractions. (So it's really cheap for what you get.)

5. With that as a basis you can now see how the higher cardinals arise.

Understanding the contrast between sets that can be enumerated versus those like the real numbers that cannot, gives you the concept of "enumerability", not just enumeration. But the concept of an ordinal indexing of an infinite set regarded abstractly as a list in order to keep track of its elements is not the same concept as comparisons of "infinities" somehow regarded as the "size" of infinite sets in which these "sizes" are ordered as the cardinals. The concept of enumerablity as an indexing is only the starting point. More is required to understand the meaning of the comparison of sets using infinite cardinals (which was one of Tom's basic questions that we now finally get to).

The comparison of cardinals is based on the idea of the ever-increasing combinatoric complexity: The power set of the integers is "bigger" than the set of integers, and the power set of that set is even "bigger", and so on -- all in the sense of "bigger" based on the impossibility of a bijection because of the "2^n" combinatorial complexity of the power set.

So the same process is used to show how infinite sets whose elements are infinite sets of real numbers -- i.e. infinite sets of infinite sets -- can have an even "higher order" infinity, i.e., a higher cardinal, and so on for higher and higher levels of infinite cardinals.

That is how you meaningfully get 2^aleph = the next aleph endlessly. It is how you compare the "sizes" of infinite sets, is why there is a whole hierarchy of cardinals beyond the basic distinction of enumerability, and is the basis for the general concept of "equinumerosity" between infinite sets using bijections.

Cantor thought that this progression of infinite cardinals consisted of separate, actual entities like a Platonic view of numbers elevated to a hierarchy of infinities regarded as real; in fact the abstract meaning of these "levels of infinities" lies solely in the complexity of the combinatorics of finite processes applied to infinite sets. The infinite cardinals are a subcategory of the concept of the mathematical infinite, based on the mechanisms of their systematic formation through finite means with increasing levels of complexity.

6. But why do any of this?

With the same approach used to show that the rationals and Cartesian products are enumerable, you can show that the so-called algebraic irrationals -- irrational numbers like sqrt(2) that are roots of algebraic polynomials with a finite number of rational coefficients -- are also enumerable, even though all of the irrationals are not. The structure of the progression of the algebraic irrationals is systematic enough to limit them to being enumerable, just as are the infinity of rationals.

Yet the non-denumerability of the completely open-ended, general real numbers, are not enumerable, so aside from being a mathematical curiosity, the theory proves indirectly the existence in principle of non-algebraic irrationals -- the so-called transcendentals, numbers like pi, e and gamma -- and that there must be "infinitely more" transcendentals than the enumerable algebraic irrationals. (And as we know, there are facts that give rise to transcendental numbers, they are not subjectively superfluous.)

You can take this to the next cardinal level when you apply it to functions. The real-valued functions of a real number require all the possible combinations of real-valued ordinates (function values) in combination with all possible real-valued abscissas. Because of the combinatorics of the higher level abstraction of the concept of a function, you can see that there are "infinitely more" of the aleph_2 real valued functions of a real variable than the aleph_1 real numbers.

But there are not "more" continuous functions of a real number than there are real numbers because when a function is continuous its values at the rational abscissas determine the function everywhere. That limits the combinations just like the algebraic irrationals are limited. Likewise you can analyze, with diminishing returns for your investment in time, how this works out for compositions: functions of functions of a real variable, etc.

Cantor went on to device a whole "arithmetic" of infinite cardinals -- a mathematical system he devised for combining his cardinals -- which are not numbers and which do not obey the usual laws of arithmetic. But beyond the kinds of considerations described above and some other kinds of mathematical applications of the Cantor diagonal argument, I don't know of anything especially significant about his arithmetical system as such.

You can see why certain unusual-looking laws like n + aleph_0 = aleph_0, aleph_0 + aleph_n = aleph_n, etc. must be true -- if you follow the technical arguments -- but Cantor's general theory of cardinals is usually presented, at least implicitly, as a mathematical curiosity more than any mathematical theory of known scientific importance (and of course the mystical interpretations are rarely mentioned even out of historical interest; it seems that modern mathematical presentations allow for subjectivism but not mysticism).

I personally regard much of Cantor's theory of infinite cardinals, beyond a certain point, as a subjective house of cards in the name of mathematics -- although it may be interesting as a curiosity to see how the house of cards works out and that there is nothing more to it. If these are all "valid concepts", so are concepts of consciousness about cartoon characters. Cantor's house of cartoon characters (as opposed to many of his mathematical methods, including his important method of representing the real numbers as "fundamental sequences" of rationals) has yet to come to life for anything significant that I know of.

The aspects that are of significance to remember (past the final exam) are the general idea of Cantor's diagonal indexing scheme for an array, his reductio ad absurdum "diagonal argument", the general ideas of enumerability and equinumerosity, the role of the combinatorics of the power set, and of course the more general concepts of bijective, surjective and injective maps and the whole concept of the mathematical infinite and how it must always be dealt with by finitary means. (If something else ever arises from Cantor's theory that you actually need you can always go back and look it up.) And of course the whole development illustrates again how mathematics is grounded in physical reality, starting with our ability to integrate perceptual units into numbers, but that the science of mathematics and its hierarchy of abstract concepts of method cannot be understood if one remains restricted to a conception of mathematics arrested at the level of counting marbles.

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Sorry for the long delay, but this subject takes more than a casual response and I have been infinitely busy (not more than aleph_0, but it seems that way).  Stephen is correct that I have definite views on this subject and I hope to clarify them and answer Tom's questions, but I agree with Stephen's general statement, "no direct connection can be made between that process and concrete reality. However, such a direct connection is not a necessary requirement for a concept of method, and the logical basis, though not approached that way by Cantor, is neither arbitrary nor useless."

The last time I tried to explain this subject the explanation came out too terse for at least some readers and I was accused of "gobbledygook" :-(, so this one will be longer (but despite its initial appearance, still finite).  Even with that I will omit technical details, referring informally to results commonly explained in standard texts since the point of this is to answer Tom's philosophical questions, not to provide a complete exposition on the subject, and he (and others) have access to the technical references with basic results.  So this will still require a certain background knowledge and motivation to make the connections (and anyone reading this who is not familiar with the technical background may still understandably think it sounds like gobbledygook), but this time I will try to spell out the explanations and implications of the basic ideas more explicitly, including the basic role of 1-1 mappings that Stephen once wondered about for this application.

Tom, you are right to question the concepts and theory of infinite cardinals because infinite sets have no numerical "size".  Because they are infinite they are larger than any number, so if you think of "size" as an ordinary number there are no sizes of infinite sets to compare or to order.  But the concept of cardinality of infinite sets is based on a different, more abstract idea than a simple number, involving the complexity of combinatorics applied to infinite sets.

[...]

The aspects that are of significance to remember (past the final exam) are the general idea of Cantor's diagonal indexing scheme for an array, his reductio ad absurdum "diagonal argument", the general ideas of enumerability and equinumerosity, the role of the combinatorics of the power set, and of course the more general concepts of bijective, surjective and injective maps and the whole concept of the mathematical infinite and how it must always be dealt with by finitary means.  (If something else ever arises from Cantor's theory that you actually need you can always go back and look it up.)  And of course the whole development illustrates again how mathematics is grounded in physical reality, starting with our ability to integrate perceptual units into numbers, but that the science of mathematics and its hierarchy of abstract concepts of method cannot be understood if one remains restricted to a conception of mathematics arrested at the level of counting marbles.

Thank you for that very thorough response! :P It'll take me a while to "chew" these ideas. Fortunately, I did keep my excellent textbook for future reference.

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Phil Stein has pointed out privately that I misused the notation for the alephs in a way that implies commitment to a famous hypothesis in mathematics that has not been proved (and which I did not intend).

Based on the analogy with the "2^n" combinatorics of the power set I used the "power notation" aleph_1 = 2^aleph_0, etc. But the indices for the alephs are ordinarily used to refer to the ordering of the alephs, not their "exponents", with the standard notation meaning that aleph_1 is the next aleph after aleph_0 and so on as the indices increase.

This is an important distinction because it is not known if the cardinality of the real numbers is the next cardinal after that of the integers or if there are one or more others in between (and likewise for the others). It is known that there is an inequality in the form, using the standard notation: the cardinality of the integers = aleph_0 < 2^aleph_0 = the cardinality of the reals = the cardinality of the power set of the integers -- but not the equalities aleph_1 = 2^aleph_0 or the cardinality of the reals = aleph_1.

A famous conjecture that has never been proved is Cantor's famous continuum hypothesis, which has remained a famous unsolved problem in mathematics for over a century: the hypothesis that there is no cardinal between the cardinality of the integers and that of the reals. In the standard notation this is expressed by either of the equalities: cardinality of the reals = aleph_1 or aleph_1 = 2^aleph_0.

In terms of the original discussion above, the continuum hypothesis means there is no subset of the real numbers involving combinations of digits that are more complex than the rationals, but less complex than all the reals, in a way such that the subset has no 1-1 mapping to either. In my misuse of the standard notation I did not mean to imply the truth of the continuum hypothesis (and probably should have discussed it, but chose not to because the post was already so long). In fact I have not intended anything that would be regarded as mathematically controversial; the philosophy is another matter.

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Hi. This is first post on this forum. I've read very little of the content here, but this subject grabbed my attention.

The unstated assumption that bijection is the only way to judge the relative sizes of infinite sets strikes me as provincial. In the following I use a different approach.

Let us start with a simple example -- all the integers versus only the even ones. The bijection approach says the sets are the same size, since one can define a one-to-one mapping, y = 2x, that fully includes the two sets. 'x' is any integer; 'y' is an even integer. But consider a two-to-one map: y = x if x is even, y = x - 1 if x is odd. Again both sets are fully included. The x's include all the integers; the y's include all and only the even integers. Moreover, for every y there are two x's, so it's reasonable to claim there are twice as many integers as even integers.

How about the rationals versus the integers? The bijective approach says the sets are the same size, since a one-to-one map -- Cantor's diagonal argument -- can be defined between the two sets. However, one could also define an infinite-to-one mapping between the two sets. Let x be any rational. Let y be the largest integer less than or equal to x. Both sets are fully used, and each y has an infinity of x's mapped to it. Therefore, the set of y's is much smaller than the set of x's. This conclusion coheres with conceiving all the real numbers as points lying on a line. Obviously, the integers are sparse in relation to all points on the line. It also coheres with "part-whole logic". Start with all the rational numbers. Set aside part of them -- all the integers. Clearly, the part is smaller than the whole, since the part left behind is far from empty.

If the reader is unconvinced that my non-provincial approach is better than the bijective one at judging the relative sizes of infinte sets, then consider the following. Suppose someone handed you two very large containers of sand and asked, which container has more grains of sand in it? Would you try to count each grain in one container and then in the other? Probably not, for that would be foolishly inefficient. No, you would weigh the sand in each container, or somehow measure the spatial volume of sand in each, or possibly put them on a balance scale. By analogy one is wiser to take a different approach than counting to determine the relative sizes of infinite sets.

The non-provincial approach is not a panacea. It might not answer for example if some subset of the rational numbers is larger, smaller, or the same size as some subset of the reals (both subsets having infinite elements). However, I believe it is still more potent and less problematic than the simply bijective approach.

the science of mathematics and its hierarchy of abstract concepts of method cannot be understood if one remains restricted to a conception of mathematics arrested at the level of counting marbles.

I like it!

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Perhaps there's a valid way of thinking about infinite cardinalities of different sizes? Or does it depend on the invalid assumption of an actual infinity?

The set of integers is actually infinite (it is equinumerous with a proper subset of itself). Why is this invalid?

The set of real numbers can be made compact with the addition of either one or two additional elements (which we write as +infinity and -infinity). Why is this invalid?

If the axiomatic systems in which these constructs are defined or postulated is internally (logically) consistent then in what way are they invalid?

In a mathematical system restricted only to finite objects the mathematics necessary to do physics could not be formulated.

Bob Kolker

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The set of integers is actually infinite (it is equinumerous with a proper subset of itself). Why is this invalid?

The set of real numbers can be made compact with the addition of either one or two additional elements (which we write as +infinity and -infinity). Why is this invalid?

If the axiomatic systems in which these constructs are defined or postulated is internally (logically) consistent then in what way are they invalid?

Logic, in Objectivist terms, is the art of non-contradictory identification -- of reality. Other philosophies may differ, but Objectivists only play games with self-consistent nonsense for fun, but not when they are being serious.

In a mathematical system restricted only to finite objects the mathematics necessary to do physics could not be formulated.

That is why Objectivism holds that infinity, while not an actual metaphysical existent, is a valid "concept of method" just like addition, subtraction, multiplication, etc. See ItOE, Pages 17, and 48-49.

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And neither Merlin nor Bob addressed Tom's original question of whether the 'cardinals' make sense given that there is no actual infinity (as Cantor believed there was). Arbitrary 'mappings' and plunging in with out-of-context technical formulations treated as if they were floating abstractions and stolen concepts all beg the question of the conceptual meaning of mathematical ideas involving infinity. They can only be understood in terms of the finite progressions that must underly all such concepts and principles.

Logic, in Objectivist terms, is the art of non-contradictory identification -- of reality. Other philosophies may differ, but Objectivists only play games with self-consistent nonsense for fun, but not when they are being serious.

Playing with 'infinity' may sound like fun, but it can be dangerous :lol: so don't try it at home. (Betsy has a lot of such dangerous toys lying around the house :( )

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Perhaps there's a valid way of thinking about infinite cardinalities of different sizes? Or does it depend on the invalid assumption of an actual infinity?

It depends on the concepts of set and function.

Bob Kolker

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Perhaps there's a valid way of thinking about infinite cardinalities of different sizes? Or does it depend on the invalid assumption of an actual infinity?

It depends on the concepts of set and function

Sorry, but that does not answer his question, does not add to what he already knew, and in addition is misleading in its implication. Cantor used sets and functions as mechanisms, but believed that his infinities were real. Tom stated that he had covered the mathematical mechanisms in his course and that he wanted to know if the cardinals are a valid concept or if they depend on the invalid assumption of an actual infinite. Restating the names of the technical mechanisms used to 'construct' the cardinals does not answer that, and it incorrectly suggests that they are all that are required to properly understand the concept. The Forum is inhabited by people with a serious interest in Ayn Rand's ideas. Addressing questions of this nature require understanding her epistemology; they are not generally requests for rehashes of technical mathematics that can be found elsewhere, too often limited to only rationalistic manipulation of floating abstractions.

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And neither Merlin nor Bob addressed Tom's original question of whether the 'cardinals' make sense given that there is no actual infinity (as Cantor believed there was). Arbitrary 'mappings' and

The set of integers is an actual infinite set. The set of reals is an actual infinite set. It can be proved (by the well known diagonal argument) that the integers can be mapped -into- the set of reals but not -onto- the set of reals. This shows that the set of reals (an honest to Dedikind infinite set) is larger than the set of integers (an honest to Peano infinite set).

It sure makes sense to me. Also to just about everything mathematician born since 1900.

Just an aside: Do you know why the Imaginary Numbers are called imaginary? Because when people first saw them the said "stuff and nonsense!". With these numbers there were be no modern physics.

Well infinite sets are just as much a methodology as are imaginary numbers. No set theory, no theory of manifolds, no modern physics.

Bob Kolker

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Well infinite sets are just as much a methodology as are imaginary numbers.

Ayn Rand held that infinity is a valid "concept of method" but that nothing in the universe is actually infinite. That would violate the law of identity.

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Well infinite sets are just as much a methodology as are imaginary numbers.

Ayn Rand held that infinity is a valid "concept of method" but that nothing in the universe is actually infinite. That would violate the law of identity.

The great mathematician David Hilbert, and many others, have also held that there is no actual infinity. The mathematical infinite as potential was explained by Aristotle; infinity is not a quantity. As has been stated several times now, the inventor of the cardinals, Cantor, insisted that there is an actual infinity; he held that all of his cardinals were a whole infinite hierarchy of infinities. Bob's repeated assertion that the integers are "actually infinite" is an equivocation ignoring that historically well-known distinction between the 'actual' and 'potential' infinite and ignoring the question that was originally asked by Tom, who recognized the meaning of the mathematical infinite and knows that there is no "actual infinity".

Bob has yet to show he understands the question or what is required to answer it. To assert that the mechanics of the Cantor diagonal is understood by "just about every mathematician born since 1900" does not address the meaning of the concepts and the question of their conceptual validity in terms of the infinite as open-ended potential. Tom also already knew about the Cantor diagonal argument -- which had led to his question, which by itself does not answer it, and which has already been discussed here.

Nor does Bob's "honest to Dedekind, honest to Peano, no set theory, no physics, etc." rhetoric address the issue of what it can mean for an "infinity" to be "larger" than another "infinity", which was the original question.

Bob Kolker has shown a repeated pattern of mathematical statements that are of no explanatory value because they are offered as floating abstractions meaningless to the questioner; are isolated fragments with no explanation of what they are supposed to relate to and how, let alone how they are supposed answer the original question; or refer to knowledge already understood and discussed. He does this as if nothing had been said to address the original question and does not answer it himself, which he appears to not understand. This pattern has been identified before and is becoming annoying. The Forum is not 'show and tell' for mathematicians who do not understand what the discussion is about and who are so steeped in philosophical rationalism that they cannot discuss the conceptual issues raised in the terms expected on the Forum. It is such philosophical rationalism in formulating and presenting mathematics that leads to so much unnecessary conceptual confusion in the first place.

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Well infinite sets are just as much a methodology as are imaginary numbers.

Ayn Rand held that infinity is a valid "concept of method" but that nothing in the universe is actually infinite. That would violate the law of identity.

Nothing in the world is the cardinal 2 either. The cardinal 2 is the class of all sets that can be put in 1-1 correspondence with your feet. That class of sets does not actually exist in the world. It is pure abstraction. It exists only as an idea. But it sure is a concept of method. It is great for counting shoes and feet.

Bob Kolker

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Well infinite sets are just as much a methodology as are imaginary numbers.

Ayn Rand held that infinity is a valid "concept of method" but that nothing in the universe is actually infinite. That would violate the law of identity.

Nothing in the world is the cardinal 2 either. The cardinal 2 is the class of all sets that can be put in 1-1 correspondence with your feet. That class of sets does not actually exist in the world. It is pure abstraction. It exists only as an idea. But it sure is a concept of method. It is great for counting shoes and feet.

This is another equivocation on the meaning of the 'actual infinite' as it has been historically construed for centuries. Again, he does not even begin to address the issue raised by Tom, which cannot be swept under the rug by anyone interested in conceptual understanding instead of rationalistically manipulating floating abstractions and stolen concepts.

It also bastardizes the concept '2' (in the usual Frege-Russell formulation that has been discussed here on the Forum before). For those who think with their heads instead of with their feet, Ayn Rand's Introduction to Objectivist Epistemology explains the 'problem of universals' in concept formation --"what is the 'manness in man' or the 'twoness in two'. IOE includes such issues as the valid and invalid meaning of infinity and the role of reality in the concept 2, which concept one must hold before ever getting to the stage of more technical formulations of mathematical systemization. The relation of concepts of number and correspondence between 'sets' of the same numerical size presupposes conceptual understanding of numbers and cannot be substituted for it. Sets of sets are not concepts.

Bob appears to be more interested in promoting and exercising philosophical rationalism in general as a substitute for conceptual understanding of mathematical issues instead of addressing Tom's question and in opposition to Objectivist epistemology. He can believe whatever he wants to, but it raises the question of why he is here posting to the Forum at all. He has refused to engage in the actual topic of the thread, substituting unserious cracks such as 'counting feet', chronic equivocations and out of context factoids all serving as glib distractions as he trolls around jabbing at serious topics he does not appear to understand. If he has no interest in Objectivist epistemology he should move on to some other topic somewhere else.

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