31

I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminology that she introduces in this article is exceedingly helpful, as well as the very simple but often overlooked point that the concept of a "foundation of ...

27

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).
Even if ...

21

We can think of this as a game of "omega-nim;" to more precise since the game you are describing is impartial, operating under the normal play convention, and finite we have that the Sprague-Grundy Theorem applies. In other words, to every "hydra-ordinal" there is an "omega-nimber."
Already this suggests thinking of the plus signs in the hydra as being ...

13

The best reference I can think of for this is MathOverflow.
Contrary to some of the comments made above, foundational issues are today often a concern in mathematics and computer science. Contrasting foundational schemes is an activity not just limited to researchers in metamathematics or in mathematical logic. It occurs in computer science repeatedly as ...

10

Sam Sanders here, one of the authors of the paper you mention. Thanks for the nice words.
I will answer your questions based on my personal opinion.
You write:
[...] would like to know if it has impacted the reverse mathematics (RM) program, and what its significance is seen as. Could path integrals really require such powerful axioms?
First of all, ...

8

Goncharov, Harizanov, Knight and Shore investigated the Turing degrees of $\Pi^1_1$ cofinal branches (which they call "paths through $\mathcal{O}$"). They showed there is a $\Pi^1_1$ cofinal branch which does not compute $\emptyset'$, so certainly doesn't compute true arithmetic. On the other hand, H. Friedman showed there is a $\Pi^1_1$ cofinal branch ...

6

The ‘divides’ relation $\mid$ should properly be called a ‘has-a-multiple-of’ relation and defined without any reference to division as
$$ a \mid b \iff \exists c: ac = b $$
This definition implies $\forall a: a \mid 0$ (including $0 \mid 0$) and $\forall a: 0 \mid a \Longrightarrow a = 0$. And over the non-negative integers in particular, the relation ...

5

If NF is consistent, then yes Con(NF) would be one of these statements that are independent of NF. NF can interpret finite order arithmetic, so by that it would be subject to Godel incompleteness theorems. If Randall Holmes's proof of Con(NF) is correct, then NF is slightly stronger than finite order arithmetic, this means that all strong axioms of infinity ...

4

$\mathfrak{Q}$ is the countable random distributive lattice.
Emil Jeřábek has already pointed in his comments that there are only two possibilities for $\mathfrak{Q}$. Either there are no greatest element in $\mathfrak{Q}$ and it is the countable random distributive lattice. Or there is the greatest element in $\mathfrak{Q}$ and $\mathfrak{Q}$ is the ...

4

Edit I have read Fedor Pakhomov's comment above and his comment contains all points essential in my answer but in a much compressed form. Indeed, substitutions may be seen as forming a DAG, and Fedor also uses an argument from the rate of growth of iterated squaring vs. the linear length of a term in a DAG-like form. So my answer is rather an elaboration on ...

4

For the language which contains only $0$, $\mathsf{S}$ and $\times$, one reasonable way to describe a term is by the sequence of strings of esses. E.g.
\begin{align}
\mathsf{S}0 &\text{ is “one set of one ess"}\\
\mathsf{SS}0\times \mathsf{S}0 &\text{ is “one set of two esses and one set of one ess"}\\
\mathsf{SS}0\times\mathsf{S}0\times \mathsf{S}0 &...

4

(1) Is category theory the new language of mathematics, or recently the more preferred language?
Category theory has been proposed in 1940s and started taking over algebraic geometry and topology first in 1970s, and its application has only grown from there.
Whether it is the preferred language depends on which field of mathematics you are thinking about.
...

3

I'm fairly confident that there are no such derivations, and for good reason.
The paper I like on this topic is by Michael Dunn, Quantum Mathematics. He concludes
First-order Peano arithmetic formulated with quantum logic has the
same theorems as classical first-order Peano arithmetic. Distribution
for first-order arithmetical formulas is a theorem ...

3

The following answers the question as posed, but is a bit unsatisfactory since we will find a choiceless inner model.
In $V[X]$, let $F = \{ x \subseteq \omega_1 : \forall \alpha < \omega_1(x \cap \alpha \in V) \}$. Clearly $\mathcal P(\omega_1)^{V[G]} \subseteq F$. We claim that $\mathcal P(\omega_1)^{V[G]} = F$ using:
Lemma (Mitchell): For all $\...

3

You may look at Shelah's paper
``On logical sentences in PA''.
For a modern exposition of Shelah's work and an alternative example see ``Independence in Arithmetic: The Method of (L, n)-Models''

2

Ackermann set theory seems "pretty close" to having unrestricted comprehension, specifically in the form of the class and set comprehension schemas. Reinhardt proved that it is as strong as ZF (in particular, an additional replacement schema is not necessary). It's not entirely clear whether class comprehension is necessary.
I asked a question here about ...

2

Emmy Noether must fit in there somewhere. Computer Scientists always mention her when talking about the foundations for making sure that iterative and recursive algorithms terminate. Unfortunately, I don't know of any translations of her work from the German. Bibliography at https://enacademic.com/dic.nsf/enwiki/9878553 where the only relevant work that I ...

1

I think the theory presented here is equi-consistent with ZFC since it interpret Takeuti's system presented in his article: Construction of the set theory from the theory of ordinal numbers.
All axioms 1.1 - 1.17 can be captured in the extended form of second order arithmetic presented in this posting.

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