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In the article History Of Logic, more precisely in the section dealing with " Logic After 1900" one can find this assertion :

"First-order logic is not capable of expressing all the concepts and modes of reasoning used in mathematics; equinumerosity (equicardinality) and infinity, for example, cannot be expressed by its means. For this reason, the best-known work in 20th-century logic, Principia Mathematica (1910–13), by Bertrand Russell and Alfred North Whitehead, employed a version of higher-order logic."

Is it actually the case that equinumericity or infinity cannot be expressed using F.O.L. ?

That seems strange , since, in order to define equinumericity for example, one needs ( as far as I know) : the notion of a function, and the notion of bijectivity, whch apparently can be defined using FOL.

In the same way, in order to define infinity, one needs the proper subset notion and the concept of equinumericity; but it does not seem difficult to express the proper subset notion using FOL.

I guess that Hintikka is right in his asertion. So, what do I miss?

Floridus Floridi
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There's a crucial subtlety here: in what context is our putative first-order definition taking place?

The obvious definitions of infinity, equinumerosity, etc. are indeed first-order formulas - in the language of set theory. That is, there is for example a first-order $\{\in\}$-formula $\psi(x,y)$ such that for all sets $a,b$, the sentence(-with-parameters) $\psi(a,b)$ is true in the universe of sets $V$ - that is, iff $V$ $\models$ $\psi(a,b)$.

However, note that we're leveraging the power of $V$ here, regardless of how simple the sets $a$ and $b$ are. What if we try to work more conservatively? This is where we get the impossibility results. For example, per Hintikka we have:

  • There is no sentence $\theta$ such that for every structure $\mathcal{M}$, we have $\mathcal{M}\models\theta$ iff $\mathcal M$ is infinite.

  • There is no sentence $\theta$ such that for every structure $\mathcal{M}$ with unary relations $A,B$ we have $\mathcal{M}\models\theta$ iff the sets $A^\mathcal{M}$ and $B^\mathcal{M}$ have the same cardinality.

  • There is no sentence $\theta$ such that for every structure $\mathcal{M}$, we have $\mathcal{M}\models\theta$ iff $\mathcal{M}$ is uncountable.

And so on. The most common tool for proving such limitative results is the compactness theorem, but the more technical downward Löwenheim–Skolem theorem also plays an important role.

(I've specified "downward" since the upward Löwenheim–Skolem theorem is really just a compactness corollary — this is a pet peeve of mine.)

And so forth. Note that here we're not sneaking in any "extra resources" beyond the structures in question themselves.

LSpice
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Noah Schweber
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