Tagged Questions
4
votes
1answer
59 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 ...
20
votes
4answers
367 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, ...
2
votes
1answer
66 views
Materials for studying logic
I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical ...
0
votes
0answers
79 views
mathematical logic and valid ways of reasoning
I'm curious about what math (mathematical logic, metamath) says about the way we reason. This is going to be a vague question because I have not yet explored mathematical logic myself. My question: ...
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 ...
-2
votes
1answer
116 views
What branch of mathematics improves logical thinking? [closed]
So, that's the question. I dare to generalize it even wider: what branch of mathematics improves the general thinking ability, intilligence, the way the person thinks, and makes it more logical? I'm ...
7
votes
3answers
133 views
Measure of how much information is lost in an implication
In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$:
...
4
votes
3answers
194 views
Does learning logic and set theory before arithmetic, algebra, and geometry have an advantage?
I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic ...
16
votes
5answers
1k views
Is 1+1 =2 a theorem?
A theorem is defined to be a mathematical statement that is proven to be true. The statement $1+1=2$ has definitely been proven in the history of mankind (Russel and Whitehead had once proven it in ...
37
votes
13answers
3k views
Is there such a thing as proof by example (not counter example)
Is there such a logical thing as proof by example?
I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right.
This works and is ...
7
votes
1answer
90 views
Independence results that cannot be established by forcing.
I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
2
votes
1answer
115 views
Is my book a good introductory one?
I attempting to study logic by myself, I acquired the book A mathematical introduction to logic by Herbert B. Enderton, I want to ensure that this is a good introduction so, is this a good ...
1
vote
1answer
49 views
Uniform continuity on empty set.
Let $\langle X,\rho \rangle$ be a metric space and $f:\emptyset\to X$ a function. Since $\emptyset$ is compact, I know that $f$ is uniformly continuous. But can it be proven by vacuous truth? It's the ...
1
vote
4answers
128 views
Would it also be useful to include an ordered pair function in first order logic?
Typically, first-order logic is assumed to include an equality relation $=$, even though this is "non-logical," together with some postulates about equality.
Would it also be useful to include an ...
2
votes
1answer
80 views
Successor axiom systems and sequences of axiom systems
Let $A$ denote a system of first-order axioms. Is there a canonical way to form a successor system $A'$ extending the ontology of $A$ to include all definable collections?
Edit: Importantly we want ...
1
vote
1answer
41 views
How often do incomplete types meet the hypotheses of the omitting types theorem?
I find the following formulation of the hypothesis (namely, non-isolation) for the omitting types theorem. A type $p$ over $T$ is "isolated" iff there is a formula $\phi(\vec{x})$ such that $\exists ...
9
votes
4answers
311 views
How to introduce advanced set-theoretical objects to philosophy students?
First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce ...
34
votes
6answers
3k views
How do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
9
votes
1answer
410 views
Choosing a Master Thesis Topic: Logic - Model Theory
I am a first-year graduate student in maths. Around these days, I feel I must decide on which exact part of mathematics I shall go through. Infact, I have narrowed down the suitable options but still ...
9
votes
4answers
143 views
Hyperreal field extension
In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?
Is it ...
3
votes
5answers
319 views
Common sense in mathematics
Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. ...
3
votes
5answers
357 views
How to interpret material conditional and explain it to freshmen?
After studying mathematics for some time, I am still confused.
The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
2
votes
1answer
167 views
Importance of Kripke–Platek set theory
What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.
4
votes
4answers
189 views
Examples of partial functions outside recursive function theory?
My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.
In recursive function theory one studies partial functions on the set of natural numbers.
Are ...
20
votes
5answers
715 views
How is a system of axioms different from a system of beliefs?
Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
(PD: I'm not religious)
3
votes
1answer
121 views
Reference request for examples of probabilistic heuristics, help put some examples in a broader context.
I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin ...
5
votes
3answers
369 views
What are the reasons for not supporting constructive mathematics
It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting constructive stance in mathematics?
5
votes
1answer
259 views
Is there any direct application of Gödel's Theorems outside of logic?
Gödel's incompleteness theorems was a major achievement with ramifications outside the field of mathematics itself. Are there any direct applications of the theorem(s), or any of the methods pioneered ...
-3
votes
4answers
208 views
What are some primary mathematical utilities of the axiom schema of separation?
I read a discussion concerning the axiom schema of specification, which I yet take as saying that for every set and a class-defining condition, those elements of the set satisfying this condition ...
14
votes
8answers
867 views
Where to begin with foundations of mathematics
I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
7
votes
3answers
689 views
Is Foundational Research a Dead Field?
I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
3
votes
1answer
134 views
A Formal and Precise treatment of Simplification?
I am looking to gain a deeper understanding of, and increase my own skill in "Mathematical Simplification". But I've been finding the concept overly vague and haven't been able to find any good ...
0
votes
2answers
77 views
Impredicative Comprehension about classes?
The Predicative Comprehension in NBG is $\exists X\forall y(y\in X \iff \phi)$, where $\phi$ is a formula where no bound class variables occur.
A possible quantification over classes (I know this is ...
3
votes
3answers
226 views
How to start with automated theorem proving?
I'm interested in this question, but I'm not going to list my knowledge/demands but rather gear it to more general purpose; so the first thing concerns the prerequisites, i.e.
How much ...
5
votes
1answer
121 views
State of the progess of the automated proof checking
I recently came across a concept of automated proof checking. I am very intrigued by the idea, that in the future all the proofs could be verified by a computer.
Moreover, some proofs were already ...
3
votes
3answers
213 views
$\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?
In "$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?"
It has been shown that arithmetic shouldn't be included. So the new modified question is:
The analogy of $\wedge,\cap$ and ...
3
votes
2answers
158 views
$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?
Update : Should have left the Arithmetic out of this question, the new
modified question is posted here : $\wedge,\cap$ and $\vee,\cup$
between Logic and Set Theory always interchangeable?
...
4
votes
3answers
327 views
Why is Kunen inconsistency at the top of Cantor's upper attic?
Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below:
So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
11
votes
4answers
415 views
Implication and Interpretation of Banach Tarski
As I understand, the Banach-Tarski paradox says a ball in 3-space
may be decomposed into finitely many pieces and reassembled into two balls each of
the same size as the original. Despite being called ...
2
votes
2answers
130 views
A theorem about inductive inference
In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read:
'Inductive inference is well known to be a hazardous ...
4
votes
4answers
330 views
Side-stepping contradiction in the proof of ; ab = 0 then a or b is 0.
Suppose we need to show a field has no zero divisors - that is prove the title - then we head off exactly like the one common argument in the reals (unsurprisingly as they themselves are a field).
...
2
votes
0answers
99 views
Lectures of many valued logic
I am looking for a good introduction to this topic... something with lots of examples and models would be nice.
I am specially interested by the case where the truth values are open sets in a ...
3
votes
2answers
905 views
How to solve this type of Puzzles (Syllogism)?
I have seen lot of questions of below type predictions. I can't figure out the answer. Is there any common method to solve this ?
Please consider the following example
...
3
votes
2answers
187 views
Different standards for writing down expressions in a formal way
What are standard ways to write mathematical expressions in a (semi)formal way ?
In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all ...
1
vote
2answers
287 views
Confused about Wikipedia definition of NP
I've been checking my understanding of the definitions of NP and NP-complete and
I am confused by some of the definitions given on Wikipedia;
for example, the
article about NP-complete describes NP ...
4
votes
0answers
181 views
Interesting applications of the cofinite topology?
Background: I'm doing some expository writing on intuitionistic logic and I have been toying with the idea of demonstrating its applicability via models where the denotations are taken from a Heyting ...
0
votes
1answer
78 views
The resolution method
how do I found the resolvent of this formula?
$(A \vee C) \wedge ( \neg A \vee B) \wedge C$
Is it as easy as take every clause with the same statements?
$(A \vee C)$ and $( \neg A \vee B)$ and put ...
3
votes
0answers
215 views
logic lectures on youtube
Currently I am reading
Logic an Structure by Dirk van Dalen (2008).
As I am missing some basics I try to find related lectures on youtube.
I frequently watch MIT, Stanford, and University of ...
3
votes
2answers
308 views
Is it true that for all proofs of the statement, $\forall x \exists y : R(x, y)$, then we can say $y = y(x)$? (Example given)
Is it true that for all proofs $\forall x \exists y : R(x, y)$, then $y = y(x)$?
A while back I remember reading a book on functional programming that was leading into some questions about what ...
11
votes
6answers
1k views
Why do statements which appear elementary have complicated proofs?
The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if ...
