4
$\begingroup$

There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others. My question is can we describe some requirements (in some logic system) that any foundational language must satisfy?

In praticular, I have been thinking about two requirements:

  1. It can describe the logic system it lives in, for example in category theory we have the notion of a topos.

  2. It can talk about itself, this is some kind of version of large-cardinal axioms: This is going to be very vague, but please bear with me: Let $\cal{L}$ be a language (in whatever logic system we need to make this rigurous). We can extend it to include another generic object, $\cal{L}[U]$ which would serve as the universe. We require the following template of axioms on $\cal{L}[U][E]$: Every predicate in $\cal{L}$ can be extended to one on $\cal{L}[U][E]$, we reuqiure it has an interpretation in every "internal universe" $U$ and is satisfied in $\cal{L}$ iff it is satisfied in every $U$

In set theory, those $U$ are inaccessible cardinals and this axiom is both the existance of arbitrarly large inaccessible ones, and the fact that to prove a theorem in set theory, is the same as proving it in every inaccessible cardinal. So in a sense, GNB set theory satisfies these axioms.

Can we make this rigorous? Is there a logic system that we can formulate this in? Another thing to ask, are all "universes" that can internalize the logic system they are defined in equivalent?

$\endgroup$
16
  • 3
    $\begingroup$ Using Roman $L$ and then $L[U]$ and so on is a rough deviation from the notation in set theory, especially since you mix large cardinals into this. $\endgroup$
    – Asaf Karagila
    Mar 6, 2018 at 19:09
  • 5
    $\begingroup$ I would call a theory "foundational" if and only if it interprets Andreas Blass's Theory $T$: mathoverflow.net/a/90945/2126. $\endgroup$ Mar 6, 2018 at 21:19
  • 3
    $\begingroup$ The foundations of mathematics, metamathematics, and the philosophy of mathematics are largely overlapping fields, so I think that this is strongly a philosophical question as well as mathematical. As such, an answer to "What are the requirements of a foundational theory?" is going to depend on what the "correct" philosophy of mathematics is. By which I mean, if a formalist, an ante rem structuralist, and a constructivist walk into a bar, they'll disagree vehemently about what the answer to this question is. For example, there are intuitionistic objections to your requirements. $\endgroup$
    – Not_Here
    Mar 7, 2018 at 8:19
  • 6
    $\begingroup$ You say, "to prove a theorem in set theory, is the same as proving it in every inaccessible cardinal," but this is just not true. There are statements true in all such universes, for example, that are not provable in ZFC, which many like to take as the basic set theory. And if your set theory includes the statement that there are many inaccessible cardinals, then this statement itself will not generally be true in all such inaccessible cardinal universes. And Gödel-Bernays set theory (and even Kelley-Morse set theory, which is stronger) does not prove the existence of inaccessible cardinals. $\endgroup$ Mar 11, 2018 at 23:00
  • 5
    $\begingroup$ Rather, the situation in set theory is that we have an enormous hierarchy of foundational theories, stratified by consistency strength, such as the large cardinal hierarchy, and one often seeks to measure the strength of a mathematical result by fitting it into this hierarchy. $\endgroup$ Mar 11, 2018 at 23:59

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.