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7
votes
11answers
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
4answers
489 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
1answer
308 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
1answer
305 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 ...
3
votes
0answers
421 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
2answers
702 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) ...
7
votes
1answer
565 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 ...
6
votes
8answers
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 ...
15
votes
3answers
788 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
1answer
308 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
1answer
670 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
1answer
334 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
1answer
228 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 ...
16
votes
0answers
356 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
1answer
534 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
2answers
442 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
5answers
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 ...
5
votes
5answers
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
0answers
388 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
1answer
1k 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 ...
8
votes
1answer
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
0answers
945 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
2answers
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
3answers
707 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
1answer
865 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
2answers
522 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 ...
2
votes
2answers
665 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 ...
3
votes
1answer
374 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
4answers
541 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
1answer
896 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
3answers
1k 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
1answer
252 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
2answers
554 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 ...
8
votes
10answers
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
3answers
1k 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
1answer
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
6answers
883 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 ...
15
votes
4answers
3k 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 ...
30
votes
6answers
4k views

Why hasn't mereology suceeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...
7
votes
3answers
902 views

Kunen's use of Countable Transitive Models

Hi, I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...
9
votes
2answers
591 views

Consistent hierarchy of axiomatic systems

First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight. I just learned in Sergey Melikhov's answer to another question ...
4
votes
5answers
1k views

Concrete models of abstract structures

Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set ...
4
votes
4answers
940 views

Subsystems of Peano arithmetic and incompleteness theorem

I think everyone is familiar with Goedel's incompleteness theorems. In particular they imply that PA (Peano arithmetic) can not prove its own consistency. Now my question is what is the largest ...
19
votes
1answer
1k views

Nontrivial circular arguments?

There is a famous circular argument for the Prime Number Theorem (PNT). It turns out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates that taken together imply ...
2
votes
0answers
145 views

Multitype approaches to choice?

I wonder if anyone has developed a set theory which approaches the issue of the non-emptiness of products of non-empty sets via a hierarchy of types (comparable to how Von Neumann–Bernays–Gödel set ...