# Tagged Questions

**7**

votes

**0**answers

276 views

### Is there a theory of abuse of notation? [closed]

Is there any theory about the different ways notation can be abused and which abuses are ineliminable without complicating the notation in some essential way? We can define "abuse of notation" as any ...

**6**

votes

**2**answers

956 views

### Is Turing degree actually useful in real life? [closed]

In theoretical computer science, we classify problems according to their Turing degree. Is there any practical application of this?
Edit: Given that we cannot explicitly and mechanically understand ...

**21**

votes

**4**answers

845 views

### What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...

**7**

votes

**2**answers

191 views

### Natural $\Pi^1_2$ (or worse) classes of structures?

(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently ...

**2**

votes

**1**answer

349 views

### Hilbert style axiomatic proof or sequent Calculus?

I am puzzling with the question which of the two proof systems (Hilbert style axiomatic proofs or substructural Sequent Calculi) is the most discriminatory?
With discriminatory I mean is which proof ...

**5**

votes

**1**answer

524 views

### What are current trends/questions in algebraic logic?

What are current trends/questions in algebraic logic?I mean the research developed by Paul Halmos.
And anyone could give some reference for overview of it's history?
Also any overview of it's ...

**8**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...

**13**

votes

**1**answer

1k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**20**

votes

**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

**9**

votes

**1**answer

381 views

### New research on coding in reverse mathematics?

Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey ...

**15**

votes

**1**answer

529 views

### Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA

A very soft question, but I hope not out of order here.
In the first edition of Elliott Mendelson's classic Introduction to Mathematical Logic (1964) there is an appendix, giving a version of ...

**6**

votes

**2**answers

677 views

### Road to Solovay's Land.

In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great ...

**13**

votes

**12**answers

2k views

### Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...

**11**

votes

**2**answers

1k views

### Problems with Journal Submissions due to arXiv Submission? [closed]

Hello,
I wanted to ask people if they know of any journals that will not accept papers for submission if they have been already posted on arxiv. I am personally interested in Logic Journals, but ...

**1**

vote

**1**answer

867 views

### Shortest formal statement equivalent to the continuum hypothesis

What is the shortest formal statement you can write that is provably equivalent to the Continuum Hypothesis in ZFC?
Please use only variables and the following symbols: $\forall, ...

**1**

vote

**1**answer

356 views

### Is there a countable pseudocharacter Hausdorff spaceļ¼such that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...

**6**

votes

**5**answers

710 views

### the example of ccc but not separable

I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
...

**21**

votes

**6**answers

3k views

### What are some proofs of Godel's Theorem which are *essentially different* from the original proof?

I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by ...

**9**

votes

**2**answers

577 views

### Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem:
Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...

**0**

votes

**1**answer

642 views

### Do we need more than the periods? [closed]

Reading this question, and the Wikipedia page on reverse mathematics, I wonder whether one needs more than the subfield $\mathcal{P} \subset \mathbb{C}$ of periods for applied mathematics, or indeed ...

**5**

votes

**2**answers

1k views

### How quickly did Goedel's Incompleteness Theorem become known and heeded throughout mathematics

Does anyone know how news of Goedel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me ...

**12**

votes

**4**answers

2k views

### Is modern computability theory “really” about algorithms?

Apologies if my question seems overly naive, but I haven't seen/heard/read any good answers.
What is modern computability theory "really" about? The study of feasible(even remotely feasible) ...

**1**

vote

**4**answers

617 views

### Condition of possibility = Co-Implication

Sorry, but I do not know another place to post this question.
Condition of possibility is an important philosophical concept. Naively, this concept could be formally defined this way:
$q$ is a ...

**26**

votes

**4**answers

3k views

### How to make Ext and Tor constructive?

EDIT: This post was substantially modified with the help of the comments and answers. Thank you!
Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most ...

**7**

votes

**1**answer

494 views

### Explicit uses of alephs above 'small ones'

In a paper placed on the arXiv today Shelah references theorem 0.9 from this paper (also Shelah) that uses $\aleph_{736}$ as an upper bound. This strikes me as analogous to Skewes' number. Are there ...

**14**

votes

**2**answers

2k views

### Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...

**28**

votes

**7**answers

5k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**8**

votes

**5**answers

832 views

### The use of the word “model” in Mathematical Logic vs the same word in Natural Sciences

I have always been wondering
why the term "model" is used by mathematicians (especially in mathematical logic) in a conceptually different (even opposite) way than it is used by other scientists, ...

**3**

votes

**5**answers

12k views

### Induction vs. Strong Induction

Is there ever a practical difference between the notions induction and strong induction?
Edit: More to the point, does anything change if we take strong induction rather than induction in the Peano ...

**19**

votes

**1**answer

1k views

### Community experiences writing Lamport's structured proofs

About two years ago, I came across this paper by Lamport
http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf
on writing proofs hierarchically. It changed how I wrote ...

**5**

votes

**6**answers

5k views

### Reading materials for mathematical logic

Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject?

**7**

votes

**4**answers

1k views

### Infinite games: are they well defined?

It is just my curiosity about this question where we have an infinite game and (according to the answers) winning strategies for both players. I am familiar with terminating games only, and I am ...

**35**

votes

**5**answers

2k views

### What's wrong with the surreals?

Of all the constructions of the reals, the construction of the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...

**18**

votes

**9**answers

4k views

### Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]

As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...

**27**

votes

**4**answers

2k views

### Is “all categorical reasoning formally contradictory”?

In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question
What was the ontological ...

**2**

votes

**2**answers

357 views

### When forcing with a poset, why do we order the poset in the order that we do?

In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it ...

**-1**

votes

**1**answer

301 views

### Properties of collections (functions) that make them proper classes (uncomputable)

There are collections too big to be a set, e.g. the collection of all sets (in ZFC), and there are collections that cannot be sets for "pure" logical reasons, e.g. the collection of sets that do not ...

**6**

votes

**1**answer

655 views

### An ubiquitous pattern of questions

There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally):
Can a ...

**4**

votes

**6**answers

884 views

### Are there nonequivalent randomnesses?

There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)
...

**7**

votes

**3**answers

3k views

### Is functional programming a branch of mathematics?

In Theory mainly concerned with lambda-calculus?, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:
That would never stick unless there's another good ...

**3**

votes

**5**answers

1k views

### Theory mainly concerned with $\lambda$-calculus?

Automata theory is mainly concerned with Turing machines and all its relatives-in-spirit. $\lambda$-calculus is rather rarely mentioned in textbooks on automata theory.
What's the common name of the ...

**22**

votes

**5**answers

2k views

### Proof assistants for mathematics

This question is related to (maybe even the same in intent as) Question 1017, but none of the answers seem to address what I'm looking for.
There are a lot of resources available for people who want ...

**2**

votes

**2**answers

690 views

### increasing bijection

Using the back-and-forth method we can construct an increasing bijection from the set of rational numbers to the set of of rational numbers except zero.
...

**9**

votes

**3**answers

952 views

### Is formal proof (formalized mathematics) interesting to practicing mathematicians? To educators? [closed]

Formalizing mathematical proofs so that they can be checked for correctness and manipulated by computer is a recurrent proposal, most notably stated in the QED manifesto (1994). The December 2008 ...