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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 ...


26

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 ...


12

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 ...


12

Colin McLarty has looked into this The large structures of Grothendieck founded on finite order arithmetic, Review of Symbolic Logic 13 issue 2 (2020) pp. 296--325, doi:10.1017/S1755020319000340, arXiv:1102.1773. with abstract (emphasis added): The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in ...


5

Let $|X| > \mathbb{N}$ and $f \in X^X$, we construct a function $g$ that commutes with $f$ but is not a power of $f$. Let $G$ be the directed graph of $f$. Split $G$ into connected components. Call a connected component boring if it is of the form $a_1, a_2, \cdots$ or $\{a_i\}_{i \in \mathbb{Z}}$ with $f(a_i) = a_{i+1}$. If $G$ has no non-boring ...


5

No, and here is a counterexample. Suppose that $|\Bbb R|<|[\Bbb R]^\omega|$, that is, there are more countable subsets of reals than reals. This is indeed possible, e.g. if all sets of Lebesgue measurable. Since $\sf ZF$ proves there are bi-surjections (in fact, an injection from $\Bbb R$ into $[\Bbb R]^\omega$), this would be a counterexample. Now, $|\...


5

Let $\kappa$ be a sufficiently large regular cardinal. Take a countable elementary substructure $H$ of $H(\kappa)$ containing $z$ and $<_\alpha$ with $\omega_1\cap H\in D$. Let $\pi : M\to H(\kappa)$ be the inverse of the transitive collapse of $H$. To see the fifth bullet point, it suffices to show that in $M$, $\pi^{-1}(<_\alpha)$ has rank less than $...


5

No such section exists. The main point in the proof is that there exist uncountably many (in fact continuum many) infinite subsets of $\mathbb N$ such that the intersection of any two of them is finite. (Such sets are called almost disjoint.) In order to preserve $\cap$ and $\emptyset$, a section would have to send the equivalence classes of these almost ...


4

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 ...


3

(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

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

I claim that a non-atomic measure $\mu$ can never be $<{2^\omega}^+$-additive. Then the same applies to any finitely-additve extension. Let $(\Omega, \frak{A}, \mu)$ be a measure space and let us assume that $\mu$ is non-atomic. It follows that there exists $A \in \frak{A}$ such that $0 < \mu(A) < \infty$. I now want to partition $A$ into $2^\omega$...


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 ...


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 ...


1

It's a bit long for a comment, but I'll make several points. These are not uncommon ordinals. I've seen them used in Rathjen's ordinal collapsing function involving Mahlo cardinals, which he denotes $\Phi$. As the comments point out, they appear in various places. This is not at all how the multivariable Veblen function behaves (before the edit). Your $\...


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.


1

The answer is negative. Suppose $(X,\mu)$ is such a measure space. By the argument in the linked question, there is a partition of $X$ into continuum-many pairwise disjoint $\mu$-null sets. This is done by building a binary tree that splits a given node into two nodes of one half the measure. Each branch corresponds to a point in Cantor space. We induce ...


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