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Cantor's general theory of sets

It was a widespread belief in the late 19th century that pure mathematics was nothing but an elaborate form of arithmetic and that the “arithmetization” of mathematics had brought about higher standards of rigor. This belief led to the idea of grounding all of pure mathematics in logic and set theory. The implementation of this idea which proceeded in two steps:

1. the establishment of a theory of real numbers (arithmetization of analysis)
2. the definition of the natural numbers and the axiomatization of arithmetic

Infinite sets had been needed for an adequate definition of important mathematical notions, such as limit and irrational numbers. It was this initial use that led Cantor himself to begin studying infinite sets “in their own right.”

In 1883, Cantor began his general theory of sets with the publication of Foundations of a General Theory of Manifolds (the Grundlagen). Among other things, in this paper he made the interesting and somewhat self-justifying claim of the autonomy of pure mathematics:

• pure mathematics may be concerned with systems of objects which have no known relation to empirical phenomena at all
• any concepts may be introduced subject only to the condition that they are free of contradiction and defined in terms of previously accepted concepts

In the Grundlagen, there was a significant change in Cantor’s conception of a set, which he defined as follows:

any multiplicity which can be thought of as one, i.e. any aggregate (inbegriff) of determinate elements that can be united into a whole by some law.

The notable changes in this from his earlier explanation of the concept of set are these:

• the absence of any reference to a prior conceptual sphere or domain from which the elements of the set are drawn
• the modification according to which the property or “law” which determines elementhood in the set “unites them into a whole”

The theory of transfinite numbers

In Cantor’s overall reason for introducing a new conception of set was to support the development of his theory of transfinite numbers.

Cantor intention was "to generalize in a rigorous way the very notion of number in itself ... by building transfinite and finite numbers, using the same principles." In order to do this, Cantor employed the notion of set in an entirely new way:

• Cantor’s previous notion of a set involved specifying a set of objects from some given domain, albeit one which was already well-defined.
• Cantor’s new notion of a set introduced the transfinite numbers in terms of the notion of a set of objects of that very same domain.

Briefly, Cantor defined (ordinal) numbers to be what can be obtained, by starting with the initial number ($0$) and applying two operations, which he called principles of generation:

1. the usual process of taking successors, which yields, for every given number $a$, its successor $a + 1$
2. a new process of taking limits of increasing sequences, which yields, after any given sequence of numbers without a last element, a number $b$

Cantor defined numbers (both finite and transfinite numbers) in terms of the notion of a class of numbers, as follows:

Let $Ω$ denote the class of all (ordinal) numbers

and $X$ range over sets

and $S(X)$ be the least number greater than every number in $X$, given by one of the two “generating principles” noted above.

and $X \text{ is a subset of } Ω ⇒ S(X) ∈ Ω$.

Then we have

• $0$ the least number is $∅$ (the null set)
• $1$ is $S(0)$ or $\{0\}$
• $2$ is $S(1)$ or $\{0, 1\}$
• $n$ is $S(n - 1)$ or $\{0, 1, 2, … n-1\}$

. . .

• $\omega$ is limit of $\{0, 1, 2, … \}$ (the set of all finite ordinals)
• $\omega + 1$ is $S(\omega)$

. . .

• $\omega_1$ (the set of all countable ordinals)

. . .

• $\omega_2$ (the set of all countable and $ℵ_1$ ordinals)

. . .

• $\omega_{\omega}$ (the set of all finite ordinals and $ℵ_k$ ordinals for non-negative integers $k$)

. . .

In order of increasing size, the (ordinal) numbers are then

$0, 1, 2, ..., \omega, \omega+1, \omega+2, ..., \omega+\omega = ω·2, …, ω·n, ω·n +1, …,ω^2, ω^2+1, …, ω^ω, … \text{ and so on and on }$ Finally, ::$Ω$ is well-ordered by $<$ (there are no infinite descending sequences of $a_n$). In 1892, Cantor proved this theorem:'"`UNIQ--ref-0000000C-QINU`"' ::given any set $S$, there exists another set, what we now call the power set $p(S)$, whose cardinality is greater then $S$ (Cantor’s Theorem). Reasoning by analogy, Cantor also argued that there is an entire infinite and very precise hierarchy of transfinite (cardinal) numbers, as follows:'"`UNIQ--ref-0000000D-QINU`"' # for a finite set of $n$ elements, its power set has exactly $2^n$ elements # the set of natural numbers $N$ has power (cardinality) $ℵ_0$ # the power set of $N$, then, has cardinality $2^{ℵ_0}$ # the power (cardinality) of $N$ is smaller than thwithe cardinality of its power set, $2^{2^{ℵ_0}}$ # and so on ... thus defining an infinite hierarchy of transfinite cardinals holding a precise greater than ordering: $ℵ_0 < 2^{ℵ_0} < 2^{2^{ℵ_0}} < … $. Cantor classified the transfinite ordinals and related them to cardinals as follows: * the “first number class” consisted of the finite ordinals, the set $N$ of natural numbers with cardinality $ℵ_0$, all of which have only a finite set of predecessors. * the “second number class” was formed by $ω$ and all numbers following it (including $ω^ω$, etc.) with cardinality $ℵ_1$, all of which have only a set of predecessors with cardinality $ℵ_0$. * the “third number class” consisted of transfinite ordinals with cardinality $ℵ_2$, all of which have only a set of predecessors with cardinality $ℵ_1$. * and so on and so on.... The transfinite ordinals thus formed the basis of a well-defined scale of increasing transfinite cardinalities:'"`UNIQ--ref-0000000E-QINU`"' Since 1878, Cantor had known that the reals $R$ formed a non-denumerable set, i.e. a set with a power higher than the naturals $N$. Now he proved this further result:'"`UNIQ--ref-0000000F-QINU`"' ::The number of elements in the set of real numbers, which he had previously termed $c$, is the same as the number of elements in the power set of the natural numbers. In other words, he proved the equation $c = 2^{ℵ_0}$ to be true, meaning that the number of points of the continuum provided by the real line had exactly $2^{ℵ_0}$ points. All of this made it possible to formulate more precisely the problem of the continuum; Cantor's conjecture became the hypothesis that $c = ℵ_1$.:'"`UNIQ--ref-00000010-QINU`"' A partial summary of Cantor’s achievements arising from his theory of transfinite numbers includes an “almost modern” exposition of the theory of [[Well-ordered set|well-ordered sets]] and also the theory of [[Cardinal number|cardinal numbers]] and [[Ordinal number|ordinal numbers]].'"`UNIQ--ref-00000011-QINU`"' In the view of some (but certainly not all) mathematicians, Cantor’s study of infinite sets and transfinite numbers introduced little that was alien to a “natural foundation” for mathematics, which “would, after all … need to talk about sets of real numbers” and “should be able to cope with one-to-one correspondences and well-orderings.”'"`UNIQ--ref-00000012-QINU`"' Here is a succinct and robust defence of what Hilbert subsequently called “Cantor’s paradise”:'"`UNIQ--ref-00000013-QINU`"' ::There are many mathematicians who will accept the ... theory of functions as developed in the 19th century, but will, if not reject, at least put aside the theory of transfinite numbers, on the grounds that it is not needed for analysis. Of course, on such grounds, one might also ask what analysis is needed for; and if the answer is basic physics, one might then ask what that is needed for. When it comes down to putting food in one’s mouth, the ‘need’ for any real mathematics becomes somewhat tenuous. Cantor started us on an intellectual journey. One can peel off at any point; but no one should make a virtue of doing so. ==='"`UNIQ--h-2--QINU`"'The paradoxes=== Cantor came to recognize that his new notion of set, which he introduced to support the development of the transfinite numbers, was problematic:'"`UNIQ--ref-00000014-QINU`"' ::not every property of numbers “unites the objects possessing it into a whole” The significance of Cantor’s general theory of sets not only for the process of mathematical rigourization generally, but also for what Hilbert would later state as his 2nd problem particularly, was this:'"`UNIQ--ref-00000015-QINU`"' ::precisely when mathematicians were celebrating that “full rigor” had been finally attained, serious problems emerged for the foundations of set theory. The “serious problems” that emerged were, of course, paradoxes. Neither the initial introduction of infinite sets by others nor their use in his early theory of sets by Cantor himself had been problematic. However, his subsequent introduction of transfinite numbers and development of transfinite arithmetic made him aware of the potential for paradoxes within set theory. Cantor is said to have attributed the source of these paradoxes to the following:'"`UNIQ--ref-00000016-QINU`"' * the use (by Frege) of an unrestricted principle of comprehension * the acceptance (by Dedekind) of arbitrary subsets of a Universal Set (''Gedankenwelt'') More specifically, the claim is that Cantor himself traced the paradoxes to a faulty understanding (by others) of what constitutes a legitimate mathematical collection. For Cantor, the mathematically relevant notion of a collection is said to have been based on the “combinatorial concept” of a set:'"`UNIQ--ref-00000017-QINU`"' ::In order to be treated as a whole, [a mathematical collection] must be capable of being counted, in a broad sense of "count" which means ''well-orderable''. In contrast to this was the “logical concept” of a set, developed by Frege, accepted by Dedekind, and championed by Russell, which “treats collections as the extensIons of concepts”:'"`UNIQ--ref-00000018-QINU`"' ::For a multiplicity to be treated as a mathematical whole, we must have some propositional function which acts as a rule for picking out all of the members. The point of this contrast rests on the claim that the set-theoretic paradoxes are only a problem for the logical concept of a set, which includes the inconsistent Comprehension Principle, and has Russell's Paradox as a result.'"`UNIQ--ref-00000019-QINU`"' The general consensus, both then and now, however, is that Cantor’s own construction of the system of transfinite numbers introduced foundational problems in the form of paradoxes into mathematics.'"`UNIQ--ref-0000001A-QINU`"' It is known that he was aware of at least two such paradoxes:'"`UNIQ--ref-0000001B-QINU`"' '''Burali-Forti Paradox''' (paradox of the ordinals) ::As Cantor defined them, each transfinite ordinal is the order type of the set of its predecessors: ::* $ω$ is the order type of $\{0, 1, 2, 3, …\}$ ::* $ω+2$ is the order type of $\{0, 1, 2, 3, …, ω, ω+1\}$ ::* and so on, so that to each initial segment of the series of ordinals, there corresponds an immediately greater ordinal. ::Now, the “whole series” of all transfinite ordinals would form a well-ordered set, and to it there would, therefore, correspond a new ordinal number, $o$, that would have to be greater than all members of the “whole series”, and in particular $o < o$. '''Cantor’s paradox''' (paradox of the alephs): ::As Cantor defined them, to each aleph is the cardinality of a class of transfinite ordinals (as described above) ::If there existed a “set of all” cardinal numbers (alephs), applying Cantor's Theorem yields a new aleph $ℵ$, such that $ℵ < ℵ$. There are disagreements concerning Cantor’s notion of number and how that notion related to the concept of a well-ordered set. There is also disagreement concerning the paradoxes of which he actually aware and when he became aware of them. There is, on the other hand, general agreement that Cantor understood the paradoxes to be “a fatal blow to the ‘logical’ approaches to sets favoured by Frege and Dedekind” and that, as a result, he attempted to put forth views that were opposed to the “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’.”'"`UNIQ--ref-0000001C-QINU`"' The paradoxes convinced both Hilbert and Dedekind that there were important doubts concerning the foundations of set theory. Cantor apparently planned to discuss the paradoxes and the problem of well-ordering in a paper that he never actually published, but the contents of which he discussed in correspondence with Dedekind and Hilbert.'"`UNIQ--ref-0000001D-QINU`"''"`UNIQ--ref-0000001E-QINU`"' Cantor did not regard the paradoxes (of which he was aware) as a crisis in set theory, but rather as a spur for the overall delimitation of sets. He considered the class of all ordinals $Ω$ and the class of all cardinals $ℵ_n$ to be “inconsistent multiplicities” and argued that whatever is deemed a set “can be well-ordered using a procedure whereby a well-ordering is defined through successive (recursive) choices:'"`UNIQ--ref-0000001F-QINU`"' ::The set must get well-ordered, else all of $Ω$ would be injectible into it, so that the set would have been an inconsistent multiplicity instead.

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References

• Curtis, G N. (1995). “The ‘Villain’ of Set Theory,” [Review of Lavine, S. (1994). Understanding the Infinite, Harvard Univ. Press], Russell: The Journal of Bertrand Russell Studies, Vol. 15, No. 1, 1995, McMaster University, URL: https://escarpmentpress.org/russelljournal/article/viewFile/1882/1908, Accessed: 2015/07/16.

• Nunez, R E. (2003). “Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals,” Journal of Pragmatics, Vol. 37, (2005), pp. 1717–1741, URL: http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf, Accessed: 2015/010/02.

• Kanamori, A. (2007). “Set Theory from Cantor to Cohen,” Handbook of the Philosophy of Science, Volume: Philosophy of Mathematics, Vol. ed: Andrew Irvine, 2007, Elsevier BV, [A revision with significant changes of “The Mathematical Development of Set Theory from Cantor to Cohen,” The Bulletin of Symbolic Logic, Vol. 2, 1996, pp. 1–71], URL:http://math.bu.edu/people/aki/16.pdf, Accessed 2015/10/02.

• Weisstein, Eric W. "Ordinal Number." From MathWorld--A Wolfram Web Resource. URL: http://mathworld.wolfram.com/OrdinalNumber.html, Accessed: 2015/10/02.