Basics of Potential Infinite, Actual Infinite and Cantor’s Theory of the Infinite

A potential infinite is a never-ending process, such as counting numbers (class notes). An actual infinite is a completed set infinite in size (class notes). Several concepts are entailed in the concept of infinity, namely, functions, one-to-one correspondence, cardinals and ordinals. A function is a set of ordered pairs such none of the ordered pairs have the same first coordinate; a function maps the first coordinate from the domain onto a second coordinate in the range (Copi, 187). For example, if F is a function containing a set of ordered pairs, and an ordered pair is <x,y> with x and y being variables, then no ordered pair within the  would have the same x. A one-to-one correspondence would be a function in which each distinct x of each ordered pair mapped onto one distinct y (Copi, 188). Cardinals are the size of any set, whereas, ordinals are the ordering of any set (class notes). In other words, ordinals are numbers representing the elements in the order of the set-the length, and cardinals are numbers which represent the number of ordinals in the set-the size (Weisstein, online; NYU-Poly, online). For example, within a set such as {0, 1, 2}, the ordinals are 0, 1, 2 whereas the cardinal number is 3; there are three numbers in the set making the size of the set three.

If two sets are the same size, if they have the same cardinality, then they would have the same one-to-one correspondence (Copi, 188). A finite set is a set onto which a one-to-one function maps the set of natural numbers less than a specific number (class notes). A finite set would not be equivalent to one of its proper subsets, however, an infinite set would be (Copi, 188). Galileo’s paradox of the infinite asserts the set of perfect squares and the set of natural numbers would be the same size; they have the same one-to-one correspondence (ibid.). Another paradox asserts all natural numbers is equivalent to all even natural numbers; the subset of natural numbers would have the same size as all the natural numbers (class notes). These two paradoxes show the subset of an infinite set is the same size, has the same one-to-one correspondence, as the infinite set which seems to show that an actual infinite is not possible because it leads to a contradiction (class notes).

Cantor argues these paradoxes are equivocating on the word ‘size’ (ibid.). An infinite set is one that can be mapped via a one-to-one correspondence with one of its proper subsets; it has the same size, but not in the sense previously conceived of (ibid.). Per Cantor, at the finite level, the cardinal number has a one-to-one correspondence with the ordinals (ibid.). At the finite level the ordinals and the cardinals line up, however, at the infinite level they split off and no longer represent a one-to-one correspondence (ibid.). In other words, the relationship between subsets of numbers at the finite level, or the relationship between the ordinals and the cardinals, works out (ibid.). However at the infinite level, the way of determining the size, i.e. cardinal, of the set of ordinals is different, and this is where Cantor introduces his transfinite numbers (class notes; Weisstein, online). At the infinite level, the words ‘size’ and ‘number’ no longer mean the same thing (class notes). What Cantor’s work arguably shows is that these paradoxes are not contradictions, it is just that numbers work differently at the finite level versus the infinite level (ibid.).

Cantor further argued that the concept of potential infinity entails the concept of actual infinity (ibid.). A potential infinity is an ever increasing variable quantity (ibid.). A variable can range over any multitude of different things (ibid.). In first order logic, the variable ranges over all objects in a domain, therefore, a domain is required for a variable (ibid.). The potential infinite would thus be the domain over which the variable ranges over (ibid.). The domain of the potential infinite would be the actual infinite, because the domain is the completed set (ibid.).

The question asked is if Cantor’s defense of the actual infinite is plausible. To be perfectly honest, I don’t know because I haven’t studied enough mathematics to be able to work through the proofs for his argument. However, at least on the surface his account of the infinite seems plausible, under the condition that one accepts his definitions of “size,” which very well could require working through the proofs in order to accept. A completed infinity can be conceived of if “size” does not mean “number.” If “size” is equated with “number,” then an infinite number would have to mean an infinite size. If so, an absolute infinite doesn’t seem possible because there does not seem to be a highest number (thus, no highest size and no completed set). Something of this sort is what the paradoxes of the actual infinite seem to have rested upon, and so if this premise is removed, then an absolute infinite may be possible.

Works Cited

Copi, Irving M. Symbolic Logic 5th Ed. Prentice Hall. 1979. (179-180; 187-188; 190)

Polytechnic Institute of New York University. “07. Cantor’s Theory of Ordinal and Cardinal Numbers.” n.d. 21 October 2012. <http://ls.poly.edu/~jbain/Cat/lectures/07.OrdsandCards.pdf&gt;.

Weisstein, Eric W. “Cardinal Number.” 2012. Wolfram Math World. 21 October 2012. <http://mathworld.wolfram.com/CardinalNumber.html&gt;.

—. “Ordinal Number.” 2012. Wolfram Math World. 21 October 2012. <http://mathworld.wolfram.com/OrdinalNumber.html&gt;.

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Basics of Cantorian Set Theory, Russell’s Paradox and Zermelo-Fraenkel Set Theory

Cantorian Set Theory assumed two axioms; the Axiom of Extensionality and the Axiom of Comprehension (class notes). The Axiom of Extensionality is: A = B iff (∀x)(x∈A ≡ X∈B) (ibid.). The Axiom of Extensionality asserts that two sets are the same if an only if all the members of one are all of the members of the other. Sets are defined by their members. The Axiom of Comprehension is: (∃S)(∀x)(x∈S ≡ φx) (Copi, 179). The Axiom of Comprehension states there is a set whose members are such that they satisfy a given propositional function. For any property given in a propositional function, there is a set; all properties have a set of objects that exists. The Axiom of Comprehension asserts what sets exist and any set that satisfies a propositional function would satisfy this axiom. The Axiom of Comprehension entails {x: x=x}; the universal set contains every property, encompasses all sets and includes itself (it is a member of itself) (class notes).

The Axiom of Comprehension and the universal set leads to Russell’s Paradox (Copi, 179; class notes). Russell found if the propositional function x∉x is input to the axiom for φx, then the propositional function is R = {x: x∉x} (Copi, 179). R satisfies the Axiom of Comprehension, so we have a set; a set such that S = {x: x∉x} which is the set of all sets that are not members of themselves (class notes). Therefore, taking x∈S ≡ φx, we have x∈S ≡ x∉x, which becomes S∈S ≡ S∉S (ibid.). The set is a member of itself if and only if it is not a member of itself. Therefore, we have a paradox. Another paradox regarding the universal set is regarding the power set (Copi, 190). The power set of a set includes all subsets of the set (class notes). Therefore, the power set is larger than the set itself (class notes; Copi, 190). The universal set is supposed to be the set that includes everything; the largest set (ibid.). But, the power set of the universal set would be larger than the universal set (ibid.). Further, the power set of the power set would be even larger (ibid.). The power set allows for larger and larger sets to go on without ever reaching a largest set (ibid.).

Zermelo-Fraenkel Set Theory attempts to resolve the paradoxes of the universal set by not assuming the Axiom of Comprehension (class notes). The Axiom of Comprehension assumes any property given in a propositional function creates a set; the sets are all out there in the universal set, we just have to pull them out of the universal set. Instead, ZF Set Theory posits the Axiom of Separation. The Axiom of Separation is: (∃S)(∀x)(x∈S ≡ x∈A · φx) (Copi, 180). The Axiom of Separation asserts if there is a set which already exists in which its members have certain property, then a new set can be created based on that property. There must be a set already in existence to which the new set is a subset of the set. The Axiom of Separation avoids Russell’s Paradox because the sets created are subsets of already existing sets which are able to not be members of themselves without contradiction (ibid.). Further, the Axiom of Separation resolves the power set contradiction because it creates sets out of already existing sets instead of asserting a largest set (Copi, 190).

All along, Cantor was trying to provide a firmer foundation for math in logic and believed set theory was a way to do this. Cantorian Set Theory with the Axiom of Extensionality and the Axiom of Comprehension was all that was needed to make set theory a foundation for math because all of Peano’s axioms of math could be derived from the combination of these two axioms (class notes). However, Russell’s Paradox and the power set paradox meant the Axiom of Comprehension had to be rejected (ibid.). Therefore, in order to make set theory a foundation for math, ZF Set Theory had to incorporate additional axioms that would be encompassing enough to be able to prove Peano’s axioms. ZF Set Theory includes the Axiom of Extensionality and the Axiom of Separation, but these two axioms were not enough to accomplish the task. Other axioms, like the Empty Set Axiom, Axiom of Pairing, Union Axiom, Power Set Axiom and the Axiom of Infinity, are needed to be able to prove Peano’s axioms.

Works Cited

Copi, Irving M. Symbolic Logic 5th Ed. Prentice Hall. 1979. (179-180; 187-188; 190)