The foundations tag has no usage guidance.

**17**

votes

**2**answers

916 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

572 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 ...

**18**

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

493 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

589 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

391 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 ...

**3**

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

208 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

228 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

358 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

175 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

516 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

347 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

328 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

445 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 ...

**4**

votes

**2**answers

763 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

605 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 ...

**7**

votes

**8**answers

2k 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

828 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

319 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

785 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

394 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

235 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 ...

**17**

votes

**0**answers

385 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

589 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

449 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 ...

**11**

votes

**5**answers

2k 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

2k 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

391 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 ...

**9**

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, ...

**6**

votes

**0**answers

964 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 ...

**3**

votes

**2**answers

1k 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

730 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

907 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

578 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

714 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
...

**4**

votes

**1**answer

406 views

### linear logic, diagrammatic calculus and foundations

Hi,
I have been interested in foundations for a while, especially categories as foundations. I am of the opinion that, as long as we present the theory of categories in SET, we will not be able to ...

**1**

vote

**4**answers

587 views

### Are inference laws consistent?

Please forgive me if this question sounds too naive... Well, in mathematics a formal theory consists of a collection of axioms $T$ (such as Peano arithmetics, or Group Theory, or ZFC), which ...

**0**

votes

**1**answer

947 views

### Is it possible to construct a finite mathematical universe? [duplicate]

Possible Duplicate:
Is there any formal foundation to ultrafinitism?
Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...

**4**

votes

**3**answers

2k views

### Dedekind's theorem

In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset. By ...

**1**

vote

**1**answer

254 views

### continuous maps between categories that are not functors

Hey,
Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? ...

**6**

votes

**2**answers

573 views

### Sets as Combinatorial Games

Just a few days ago my seemingly eternal and recurrent fascination for Conway's combinatorial game theory (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent survey, and began ...

**10**

votes

**10**answers

4k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...

**7**

votes

**3**answers

2k views

### incompleteness in real analysis

Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories ...

**15**

votes

**1**answer

1k views

### Martin's “Philosophical Issues about the Hierarchy of Sets”

Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...

**9**

votes

**6**answers

932 views

### (Non?)-linearity of the consistency strength ordering in ZF

Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...

**25**

votes

**6**answers

5k views

### How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...