The foundations tag has no usage guidance.

**9**

votes

**0**answers

678 views

### Has anyone pursued Frege's idea of numbers as second-order concepts?

Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" (...

**14**

votes

**4**answers

716 views

### Does the existence of the von Neumann hierarchy in models of Zermelo set theory with foundation imply that every set has ordinal rank?

Let $T$ be the theory consisting of Zermelo's original set theoretic axioms (extensionality, empty set, pairing, union, powerset, infinity, separation, choice) together with foundation. Put more ...

**-3**

votes

**3**answers

564 views

### Can different extensions of ZF have contradictory consequences for first-order arithmetic?

My question is basically, does there exist a statement X independent of ZF such that ZF + X implies a statement P of first-order arithmetic, but ZF + not X implies not P?
Now X cannot be the axiom ...

**11**

votes

**2**answers

598 views

### Are simplicial sets the intended model of HoTT?

While thinking about Jason Rute's question, I wondered if there was an intended model for HoTT. The main candidate for the intended model are simplicial sets, where Vladimir Voevodsky first observed ...

**17**

votes

**1**answer

682 views

### On Joyal's completeness theorem for first order logic

In 1978, in a series of unpublished conferences in Montréal, A. Joyal announced a remarkable theorem that unified several completeness theorems for fragments of first order logic, as well as first ...

**14**

votes

**3**answers

2k views

### Who introduced the terms “equivalence relation” and “equivalence class”?

Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence ...

**10**

votes

**2**answers

890 views

### Equivalent form of the Univalence Axiom

I'm reading the new HoTT book and I'm wondering about a potential equivalent form of the Univalence Axiom: $(A \simeq B) \simeq (A = B)$.
For simplicity, I'm tacitly working in a fixed universe. It ...

**11**

votes

**3**answers

2k views

### Is there a categorical proof of Gödel's incompleteness theorem?

A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new ...

**5**

votes

**1**answer

579 views

### What can be done with computability logic that previous logic systems can't?

I've been reading a lot about computability logic lately and I'm superficially aware that it unifies classical, intuitionistic and linear logics.
What I'm seeking to know is:
Can computability logic ...

**7**

votes

**1**answer

485 views

### Normality of Chaitin's constant

Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition?
Even if we replace the existential quantifiers in the assertion of non-normality ...

**5**

votes

**1**answer

726 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

**17**

votes

**2**answers

991 views

### Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here.
The question is this:
Today ...

**3**

votes

**1**answer

634 views

### Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
Part 1 is about foundations. Much of the ...

**19**

votes

**6**answers

1k views

### Where in ordinary math do we need unbounded separation and replacement?

[I have updated the question after initial comments in the hope of clarifying it.]
I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as ...

**2**

votes

**3**answers

497 views

### Are integers real? [closed]

Do you think that $\mathbb Z \subset \mathbb R$? On one hand this inclusion is quite handy. We like to write things like:
$$
\sqrt{n} \quad \text{for $n\in \mathbb Z$}
$$
which requires the number $n$ ...

**16**

votes

**1**answer

633 views

### Monte Carlo integration

As probably many other people here, I learned integration, as an undergrad, from Rudin's books. I recently realized, however, that I don't quite use Lebesgue integration in my work, or at least I use ...

**1**

vote

**1**answer

409 views

### Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...

**5**

votes

**1**answer

1k views

### Are the Foundations of Mathematical Logic Shaky? [closed]

The mathematics community at large seems pretty satisfied right now with the common practice of 1. starting with some axioms and 2. deriving theorems from them by employing some logic. All mathematics ...

**0**

votes

**1**answer

225 views

### Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...

**1**

vote

**1**answer

230 views

### Finite level super classes over ZFC

My question is: "Is it possible to have a sound and rigorous legitimation of the following construction ?". This construction is:
0/ Let ZFC be the usuel set theory, and let us add to the language ...

**24**

votes

**5**answers

3k views

### Nonstandard analysis in probability theory

I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...

**3**

votes

**1**answer

365 views

### Should functions be assumed to behave like the identity function when evaluated outside their domain?

Suppose we have a set $f$ of ordered pairs (so not a triple $(X,Y,f)$ but just the $f$) and suppose that $f$ has the appropriate property such that we can view $f$ as a function. Formally, we wish to ...

**3**

votes

**2**answers

183 views

### Dealing with undefined expressions in predicate logic

Suppose we're working in the first-order language of the real numbers, and we write
$$\forall x (x > 0 \;\rightarrow\; 1/x > 0)$$
We want this to be true, however I feel like it doesn't quite ...

**7**

votes

**11**answers

2k views

### A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...

**1**

vote

**4**answers

535 views

### Why can't an explicit well-ordering of the reals be ruled out in ZF?

The statement A = "There exists a well-ordering of the reals" is independent of ZF. My understanding is that the statement B = "There exists an explicit well-ordering of the reals" is also ...

**4**

votes

**1**answer

362 views

### weakening naive comprehension to avoid the paradoxes

Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because no consistent weakenings seem to be particularly well motivated or even to ...

**1**

vote

**1**answer

341 views

### Tarski-Grothendieck set theory, the axiom of pairing and the axiom of specification

I am building upon MO question 102846 concerning the Tarski-Grothendieck set theory (TG).
I have two questions;
1/ I think that it is possible that the axiom of pairing (axiom 4 of the TG theory ...

**4**

votes

**0**answers

463 views

### sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...

**5**

votes

**2**answers

830 views

### Large cardinals without the ambient set theory?

In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...

**6**

votes

**1**answer

613 views

### Untyped Higher Category Theory

I am currently trying to wade through the vast lake of higher category theory, a formidable task,or so it seems.
In the process, it has occurred to me that there is a basic analogy in place with ...

**8**

votes

**8**answers

3k views

### ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...

**16**

votes

**3**answers

852 views

### Finite versions of Godel' s incompleteness

Assume you have some notion of proof complexity: for instance, at the basic level, the length of a proof, or the number of symbols used, take your pick (there are more involved measures, but for sake ...

**3**

votes

**1**answer

327 views

### Godel 's Ladder: Undecidable PI_N sentences for N =2, 3, …

After Godel's groundbreaking results, a plethora of $\Pi_1^0$ undecidable arithmetical sentences have been found by many authors.
But what about $\Pi_n^0$ for $n=2,3,.....$ ?
There are, to my ...

**11**

votes

**1**answer

844 views

### Elementary Equivalence =? Homotopy Equivalence

One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here).
Finally homotopy theory ideas have entered in a royal fashion the ...

**9**

votes

**1**answer

425 views

### Ultimate Maximality Principle

I wonder if it's possible to formulate an "ultimate" maximality principle (UMP) and prove its consistency. I envision UMP to express the idea that no matter how we enlarge the universe of set theory V ...

**2**

votes

**1**answer

238 views

### comprehension and ideal elements

A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations....

**17**

votes

**0**answers

396 views

### Are there lightweight foundations for arbitrarily extendable objects?

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...

**5**

votes

**1**answer

615 views

### Some questions about Ackermann set theory

In a comment on this site Andreas Blass stated:
"To fit this situation into my philosophical point of view, I'd say that what Ackermann's theory
calls proper classes are really certain sets. That ...

**3**

votes

**2**answers

452 views

### Evolution of the Mapping/Function Concept

Hello! I'm looking for a survey (of the history) of the concept of mapping/function. How the concept was evolving. Especially I'm interested in what it turned into during the last 50 years.
So ...

**12**

votes

**5**answers

3k views

### getting rid of existential quantifiers

It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and ...

**4**

votes

**5**answers

3k views

### Easy and Hard problems in Mathematics [closed]

Modified question:
I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...

**1**

vote

**0**answers

393 views

### a priori grounds of mathematics [closed]

Hi,
from the title of the question you may guess I'm new to the site, and I am. Even though I've read the FAQ and I know that this isn't the place for such open questions (I believe it is open, but ...

**10**

votes

**1**answer

2k views

### Up-to-date version of Principia Mathematica?

Background: I found this interesting translation of Godel's On formally undecidable propositions of Principia Mathematica and related systems I that, along with translating it into English, uses more ...

**10**

votes

**1**answer

1k views

### Set-theoretical multiverse and foundations

I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, ...

**7**

votes

**0**answers

974 views

### Is there a finite-dimensional vector space whose dimension cannot be found? [closed]

Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...

**4**

votes

**2**answers

2k views

### Foundations: Existence of uncountable ordinals.

This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...

**9**

votes

**3**answers

747 views

### On a weak choice principle

[PLEASE SEE EDITS AT BOTTOM OF QUESTION]
Consider the following set-theoretic axiom:
For each set $X$ there exists a set-indexed collection $\{C_i \to X\}_{i\in I_X}$ of surjections such that for ...

**6**

votes

**1**answer

925 views

### How are mathematical objects defined from an ultrafinitist perspective?

I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...

**1**

vote

**2**answers

608 views

### Intension vs. Extension: Coextensive relations in model and set theory

(originally posted at MSE as Same same but different: Coextensive relations in model and set theory, slightly modified)
The official definition of a structure in model theory in its presumably most ...

**3**

votes

**2**answers

753 views

### products in a category without reference to objects or sources and targets

Hi,
I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape
A->B
...