Would Frege's version of the empty set contain "parafinitesimal elements," at least from the multiversal standpoint?

Question

Frege's definition of the empty set was not a raw extensional one: he did not simply write the partial string {} and say, "That's it: that's the empty set." His account was more intensional: "There is a set that contains all and only true contradictions," but since per his LNC-compliant logic he would say that there are no true contradictions, it follows that the set in question is empty (has no elements that we actually quantify over, here).

Frege's definition has been generalized (or at least adapted) to other qualifications: one that I saw was, "The empty set is the set of all ill-founded sets," but since modulo ZFC there is a "fact" that V = WF (for WF = the class of well-founded sets), such a set would be empty. So another could be, "There is a set of all disordered sets," or, "There is a set of all sets that don't have powersets," etc. Inside a world where the axioms say that all sets have property f, that world's empty set will be the set of all sets that don't have f.

But so from outside of any world as such, things might appear quite different. In the literature, they usually frame transworld witness so that the emphasis is on how elements of V can vary from world to world: some versions of V have more or different elements than other versions, V can be extended by forcing, etc. and it is variation of content on this relatively maximal level that is the concern of theorists. However, per the Fregean option, we can also look at how each world's empty set (as opposed to its universal set/proper class) has more or different elements between worlds. If V = L, for example, then 0^♯ "doesn't exist," neither do zero dagger, zero pistol, or any zero section that might be proposed. So in the constructible universe, the empty set = zero = the set of all the zero sections, which is empty only in constructible or sufficiently similar worlds but is teeming with elements from the perspective of other worlds.

The question: but suppose we were trying to be inclusive of finitism or even ultrafinitism, or at least respectful of their degree of caution with respect to infinitary reasoning, and so we wanted to introduce infinitesimals in such a way that they did not have to be reciprocals of actually-infinite numbers. Bassler[16] uses the perspicuous term parafinite to navigate these waters, so let us speak of parafinitesimals instead. Now, the relative elements of a Frege-flavored empty set could often by styled as parts/aspects/etc. of emptiness or the number zero, which despite its elementarity is fraught with nontrivial effects (e.g. the indeterminacy and undefinability of divison by zero, or the divergence between the Conway argument for 0⁰ = 1 and the quasi-model where the expression is undefined). Would describing these "parts"/"aspects" as parafinitesimally small, as in parafinitesimally close to zero without necessarily being equal/identical to it, be a way to introduce "logical infinitesimals" without properly offending finitistic sensibilities? After all, even by the strict Fregean introduction, we have that infinitesimals as examples of "actual infinity" are then nonexistent elements of an empty set in a finitistic world where the closest counterpart to "actual infinity" is the class of all sets (which following the resolutions to Russell's paradox, is usually not taken for a set, even by non-finitist set theorists). Without claiming that there are infinitely many axioms/models buttressing the appeal to the multiversal standpoint, can a finitist allow for a finite number of universes (perhaps even some where actual infinity is countenanced), such that as transworld witnesses they can see into parafinitesimal or even "properly" infinitesimal elements of an empty set?

Answer 1

You raise some intriguing technical points about the metaphysical status of infinitesimals and empty sets from a multiversal finitistic perspective. However, I think there are a few conceptual issues worth clarifying first:

• Frege defined the empty set extensionally as the set with no elements, not intensionally as "the set of all contradictions" which could imply it has contradictory elements.

• Infinitesimals are generally mathematical entities defined relative to a particular formal system or model, not objects that meaningfully "exist" in an absolute metaphysical sense across worlds.

• Multiverse theories in physics and philosophy are highly speculative and contentious. Invoking them may obfuscate rather than clarify the core issues.

• Ultrafinitism rejects even relatively small infinite numbers, so positing infinitesimals may still be objectionable from that stance.

My inclination would be to avoid metaphysically loading mathematical entities with too much "transworld existence" baggage in the first place. Formal systems are abstractions created to model certain aspects of reality, not fully describe its ultimate nature.

Perhaps a more straightforward way forward is focusing the discussion on which formal frameworks are most relevant and useful for different mathematical purposes from a finitistic perspective, without getting into metaphysical disputes over the ontological status of their entities across possible worlds. This could lead to more fruitful engagement.

In summary, while technical details matter, the bigger issue seems to be mutual understanding and bridging divergent philosophical assumptions. Establishing common ground on appropriate standards of rigor may be more productive than debates over the precise metaphysics of empty sets. But I'm happy to discuss further if you feel I'm missing key points!

Answer 2

This is a linguistics problem. If we are defining infinite expressions using finite indicators (like π, α) then we cannot describe this as proof of infinity. Until now, neither infinite nor infinitesimals have been defined. But you can coverage your expressions with "{" and "}" characters.