This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).
5
votes
3answers
107 views
There is concept of finite sets that can have only one “interpretation”?
In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
3
votes
2answers
69 views
Axiom of infinity exceptions?
It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
8
votes
0answers
113 views
Hao Wang's $\mathfrak S$ system: a “transfinite type” theory?
Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with ...
1
vote
0answers
36 views
Are there modern books (or otherwise) with very strong opinions about the correct way of setting up mathematics?
Euclid had very definite ideas about how to set up mathematics - his method involved axioms, definitions, theorems, and proofs.
Similarly, Bourbaki believed in - among other things - proceeding from ...
4
votes
1answer
60 views
Can we use proper classes in this way, to define a new infinity larger than |Ord|?
I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
1
vote
0answers
36 views
How does the Tarski axiom relate to the Grothendieck universe?
It's claimed, e.g. in the formulation of Tarski–Grothendieck set theory, that the Tarski axiom implies that for every set, there is a Grothendieck universe containing it.
However, I can't see how ...
20
votes
4answers
377 views
Is $\mathbb{N}$ impossible to pin down?
I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.
In ZFC, ...
1
vote
1answer
450 views
Philosophy of a Mathematician.
Introduction
I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects.
But I love to find structures and ...
-2
votes
1answer
175 views
What does it mean that a number exists? [closed]
The word exists is a very culturally loaded word. Yet mathematicians use it all the time. My answer proposing that the statement that there "exist a number Q such that Q*Q=2", only means that we can ...
1
vote
0answers
46 views
Can any axiom of a first order mathematical theory be written as a definition?
I have seen different axiomatizations of PA. I some, equality is defined in others is an axiom. The same can be said of addition and multiplication. So it is not clear to me why and when axioms are ...
3
votes
2answers
42 views
Show that $\exists A \subset \mathbb{R}$ such that $\forall x$ $\in \mathbb{R}$, we may write $x$ uniquely as $x=a+q$, where $a\in A,q\in\mathbb{Q}$.
Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set A. Once we've got that, it'd be a matter of ...
0
votes
0answers
29 views
I can't quite understand the proof given by E. Landau of theorem 4 chapter 1 in his “Foundations of Analysis”:
http://www.4shared.com/file/iH_AfbdC/Doc4.html here's the proof; is there anybody who would be so kind to help me understand it?
8
votes
2answers
132 views
Are there concepts of mathematics today that aren't on a solid logical foundation?
In this answer, Arturo wrote:
I would think like Sebastian, that most "working mathematicians" didn't worry too much about Russell's paradox; much like they didn't worry too much about the fact ...
1
vote
2answers
57 views
Language, set and sentential calculus
I'm trying to learn sentential calculus now and I'm very confused with the following thing.
In many books on logic I've found out that before working with sentential calculus itself we need the ...
4
votes
2answers
147 views
Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?
I've heard people make the argument that:
$\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
5
votes
1answer
80 views
Basic question about encoding ZFC into PA
1) Are ZFC and PA arithmetic mutually interpretable if we extend PA to PA+A , where A is the set formulas of PA that result from the translation of the axioms of ZFC (or any large cardinal axioms ...
2
votes
1answer
35 views
Infinite axiom schemas - what comes before?
The ZFC axioms constitute an infinite set. So too do the PA axioms. Presumably, then, we need a metatheory before we can even define these axiom systems.
What is the usual metatheory?
Do we even ...
1
vote
0answers
32 views
Using definitions instead of axioms.
Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms.
Given all this, a rigorous work of mathematics will typically begin with a signature - ...
22
votes
7answers
1k views
Does mathematics require axioms?
I just read this whole article:
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
which is also discussed over here:
Infinite sets don't exist!?
However, the paragraph which I found most ...
10
votes
1answer
112 views
Any branch of math can be expressed within set theory, is the reverse true?
Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property?
I am asking ...
2
votes
1answer
76 views
Intuitionism - is it fundamentally different than “ordinary” mathematics.
I have recently had a conversation with a person who considered intuitionism to be a valid alternative for the "usual" kind of mathematics. Clearly, intuitionism differs from the type of mathematical ...
3
votes
2answers
85 views
Are axioms and rules of inference interchangeable?
There is an equivalence between cellular automata and formal systems, you can code one into the other and vice versa. But in the the cellular automata (CA) the rules of inference are fixed and are ...
10
votes
1answer
219 views
What of the “Sets, Classes, and Categories” approach to the foundations?
A 2001 paper by F.A. Muller proposes "ARC" (a modified form of Ackermann set theory) as a foundations of mathematics, and argues that it founds category theory more naturally and/or conveniently than ...
3
votes
2answers
49 views
Which concept is more primary in modern formal math: set, function or natural number?
My reasons to ask this question are the following:
1) formal definition of a function is "a subset of cartesian product of two sets" so it is defined with the set and natrual number as basic ...
3
votes
1answer
68 views
Hilbert's Program
I'm trying to understand David Hilbert's intention in creating a mathematics foundation.
Hilbert's program intention to confirm that we can find a finite set of axioms with ...
7
votes
2answers
158 views
Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?
My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
1
vote
1answer
52 views
Axioms of arithmetic and isomorphism
What does it mean to say that the axioms of arithmetic (Peano's or Dedekind's) are sufficient to characterize the abstract, mathematical structure of the natural numbers up to isomorphism?
2
votes
2answers
129 views
Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics
As I understand that there are at least two fundamental limits of the development of the mathematics:
1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...
5
votes
0answers
72 views
Functions and metafunctions
I didn't get any responses to this question the first time around, so I've tried rewriting parts of it. If there's anything glaringly wrong with the questions I'm asking, please leave a comment!
...
30
votes
9answers
2k views
Infinite sets don't exist!?
Has anyone read this article? Set theory
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his ...
9
votes
0answers
121 views
What lessons have mathematicians drawn from the existence of non-standard models?
So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
2
votes
2answers
131 views
Hard proof concerning the periodicity of trigonometrical functions. Is that a challenge or just trivial
i want to know if exist or if you can develop or give me ideas of a proof to show that the least number for which sine is periodic is $2\pi$
$$\neg \{\exists n\in \mathbb{R} \wedge n < 2\pi: ...
6
votes
2answers
84 views
Is there a way to axiomatize the category of sets and relations?
The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?
16
votes
1answer
365 views
Are the real numbers ever needed to prove a property of the natural numbers?
Suppose no one had invented/discovered the real numbers yet (so e.g., no calculus), would this constrain the possible theorems or knowledge we could have about the natural numbers?
0
votes
1answer
46 views
Is there a systematic account of the number systems in the following 3x3 grid?
Consider the following sets of numbers, viewed as number systems with signature $(+,\times,\leq)$.
Let $\mathbb{X} = \{1,2,3,\cdots\}$ denote the nonzero natural numbers. Let the completion of ...
6
votes
0answers
169 views
Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.
Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
5
votes
3answers
117 views
Mathematical Limitations of Computer Experiments
One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
3
votes
0answers
106 views
What do ultrafinitists think about Graham's number?
I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time).
So I wonder about something like the famous ...
5
votes
2answers
104 views
Can Arithmetic recreate the transinfinite hierarchy of Set Theory?
I asked this question in Philosophy.StackExchange whilst trying to get to grips on Badious declared philosophy on using mathematics as ontology. But was advised to ask it here because of the ...
2
votes
1answer
66 views
The definition of distance and how to prove the ruler postulate in Euclidean geometry
I have started to read some books about geometry. At the moment I've just started to read Hilbert's axioms and also some elementary books for highschool. From the basic perspective of an axiomatic ...
10
votes
3answers
312 views
When we say, “ZFC can found most of mathematics,” what do we really mean?
ZFC works as a foundation because it can prove many sentences that are "translations" of theorems from "standard" mathematics into the language of ZFC.
But there's a subtlety. When we say, "ZFC can ...
1
vote
4answers
223 views
Can we modify ETCS to handle structures directly, as objects in their own right?
I have completely rewritten this question; thus, some of the comments/answers may no longer be relevant.
The elementary theory of the category of sets (hereafter, ETCS) is an axiomatic approach to ...
17
votes
2answers
705 views
How can there be alternatives for the foundations of mathematics?
How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of ...
