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* 5^{th} 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>.

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

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