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Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" (Die Grundlagen der Arithmetik) and "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetik).

(Page numbers and section numbers refer to the excerpt of the Grundlagen here.)

  1. He makes the thesis that an ascription of number is an assertion about a concept. (Concepts are what we today call properties.) For instance, when we say that Jupiter has four moons, we aren't saying something about each of its moons, like "each moon of Jupiter is four", which would be nonsense. Rather we're making an assertion about the concept "moon of Jupiter". Specifically, we're asserting that the number four applies to the concept "moon Jupiter". (§45 - §54, pg 58 - 67)

  2. He then gives an analysis of what it means for any particular number to apply to a concept. He does so recursively: 0 applies to the concept F if no object falls under F. (In modern terminology, we would say "the property F doesn't apply to anything.") For any natural number n, n + 1 applies to F if for any a that falls under F, n applies to the concept "falling under F bot not being equal to a". (§55, page 67)

  3. He then suggests that we could simply define numbers as second-order concepts using the analyses from step 2, identifying 0 with the concept "0 applies to", 1 with the concept "1 applies to", etc., as defined in 2. But he rejects this definition for two reasons: first, with this definition numbers wouldn't be objects, and he makes vociferous arguments that they are objects. Second of all, this definition doesn't allow us to talk about numerical equality: if m applies to F and n applies to G, our definition gives no criteria for when m = n. (§56 - §61, page 67-72)

  4. He solves this numerical equality problem by giving the conditions under which the number for two different concepts are equal: the number that applies to F is equal to the number that applies to G if and only if F is equinumerous with G. In other words, F and G have the same number of elements precisely when there is a one-to-one correspondence between F and G. This is known as Hume's principle. (§62 - §65, pg 73- 77)

  5. He then suggests that we could define numbers as second-order concepts using the analysis in step 4: for any concept F, the number of F would be the second-order concept "equinumerous with F". Then obviously F is equinemerous with G if and only if "equinumerous with F" applies to the same concepts as "equinumerous with G", so Hume's principle is satisfiedq. But he rejects this definition for two reasons: this also won't let numbers be objects, and also he thinks it's wrong to think of concepts as extensional rather than intensional. (§69, page 80, footnote 1)

  6. He remedies the intensional/extensional issue by defining the number of F as the extension of the concept "equinumerous with F". (Extension is what we would today call the set of all elements.) And then he axiomatizes extensions (in his next book) using his disastrous Basic Law V, what we call naive set theory: the extension of F is equal to the extension of G if and only if the same elements fall under F and G. (§66 - §69, page 77-81)

Now Russell's Paradox showed that Basic Law V led to an inconsistency, so Frege's original logicist project was a failure. But there have been various attempts to save it; John Burgess even wrote a book "Fixing Frege" detailing these attempts. Crispin Wright and George Boolos adopted Hume's principle in step 4 as an axiom and showed that it can be used to prove all the axioms of Peano arithmetic, part of an an endeavor known as neologicism. Richard Heck tweaked the logic of second-order logic so that Basic Law V in step 6 would no longer give an inconsistency. George Boolos tweaked Basic Law V itself in step 6 so it wouldn't give an inconsistency.

And most important for our purposes, Harold Hodes, in his paper "Logicism and the Ontological Commitments of Arithmetic", and Augustin Rayo, in his paper "Frege's Unofficial Arithmetic" adopted Frege's rejected attempt to define numbers as second-order concepts along the lines of step 3.

My question is, has anyone tried to pursue Frege's other rejected attempt, his definition of numbers as second-order concepts in step 5? Has anyone investigated how much of arithmetic we can reconstruct if we can define the number of F as the concept "equinumerous with F"?

Any help would be greatly appreciated.

Thank You in Advance.

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Are the interpretations 1-6 of Frege yours or do you have references for them? – François G. Dorais Sep 15 '13 at 4:02
@FrançoisG.Dorais I'm not really doing any interpretation here. I'm just summarizing what Frege says in the Grundlagen in a pretty uncontroversial manner. The only "interpretive" work, if you want to call it that, is how I chose to break his argument into six parts, for ease of reference. If you'd like, I could link to a PDF of Frege's book and note what sections of the Grundlagen correspond to each of my steps. – Keshav Srinivasan Sep 15 '13 at 4:19
Keshav, could you explain what it would mean to pursue Frege's concept further, as distinct from the widespread treatment of equinumerosity classes as a concept of cardinality and hence of number. Without AC, for example, our current main cardinality concept is just equinumerosity classes, and these are commonly referred to as "cardinals", although this usage is filtered by the recognition that they form proper classes. But did you envision something going much beyond that? – Joel David Hamkins Sep 15 '13 at 17:12
@JoelDavidHamkins What you're talking about is the use of equinumerosity classes for numbers in the context of axiomatic set theory. What I'm talking about is the development of numbers in the context of pure second-order logic, where there's no such thing as sets. See the papers by Hodes and Rayo that I linked to for one development of numbers in 2nd-order logic. (Except it uses the definition suggested in step 3 instead of the equinumerosity definition suggested in step 5.) – Keshav Srinivasan Sep 15 '13 at 18:38
If your logic is limited to first- and second-order concepts, then I don't see how you can define anything equivalent to N (the predicate satisfied by the natural numbers) if numbers are second-order, so you won't have induction. If you allow third-order concepts (and have comprehension to state their existence), then I imagine the usual Frege Arithmetic approach would go through, where second-order concepts in FA are replaced by third-order concepts in your logic. – abo Sep 16 '13 at 19:55

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