6
votes
1answer
341 views

Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is: Question 1. Which different approaches to set theoretic forcing are ...
6
votes
0answers
223 views

Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal. At most papers ...
5
votes
2answers
664 views

Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context? We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...
10
votes
1answer
302 views

Elements of the method of forcing in some papers of N. N. Luzin

In the paper Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...
8
votes
0answers
244 views

From Frege to Gödel - German equivalent?

I know this question does not quite fit here, but I felt it could best be answered here. I recently stumbled upon the book From Frege to Gödel, which is a sourcebook containing some of the most ...
2
votes
1answer
154 views

why the difference between terms and propositional variables?

Reading some old logic texts (written around 1930) I noticed that these texts make no difference between propositional variables and terms. They do make difference between identity and truthvalue ...
9
votes
1answer
453 views

Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$ whose validity is the ...
12
votes
6answers
1k views

The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory. Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...
7
votes
0answers
449 views

Has anyone pursued Frege's idea of numbers as second-order concepts?

Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" ...
6
votes
1answer
624 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. Could anyone give some references for the overview of its history? Any overview of its application ...
21
votes
2answers
1k views

Hahn's Embedding Theorem and the oldest open question in set theory

Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper Über die nichtarchimedischen Grössensysteme (Sitzungsberichte der ...
14
votes
1answer
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 ...
18
votes
9answers
3k views

Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY. George Berkeley wrote in 1734 with reference to the early calculus that ...
21
votes
2answers
1k views

Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...
15
votes
1answer
616 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 ...
35
votes
4answers
2k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
30
votes
10answers
3k views

Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...
6
votes
1answer
427 views

Who introduced the concept of Primitive recursive functions?

I have thought that Gödel introduced the concept of Primitive recursive functions in his seminal paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (I hope I ...
4
votes
1answer
227 views

History of provably total functions of a theory

Provably total functions of an arithmetical theory is one of the tools used in proof theoretic analysis of theories. I am looking for early history of its development. In particular, Where was ...
4
votes
3answers
355 views

Official names for specific compound sentences

This question is, admittedly, a little less mathematical than what I normal ask. I seemed to remember that the compound sentence $A\wedge \neg A$ has an official name (maybe even "contradiction" but ...
10
votes
1answer
630 views

What is the etymology of model?

What is the etymology of model? The answer is of course pre-WWW, but the better part of an hour in the library searching both classic model theory and modal logic textbooks turned up nothing. Every ...
5
votes
1answer
238 views

Under what conditions does $\mathcal{M} \vDash \mathsf{PA}$ and $\mathcal{K} \vDash \mathsf{PA}$ such that $\mathcal{M} \ncong \mathcal{K}$?

I'm currently learning some introductory model theory from Marker's "Model Theory: An Introduction", Kaye's "Models of Peano Arithmetic", and Hodges' "Model Theory", and I am confused by the Wikipedia ...
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 ...
12
votes
4answers
1k views

Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

In the Grundgesetze der Arithmetik, Frege used a number of strange characters for notation. I would be most interested to know anything about the typography (origin, usage and so on) of the strange U ...
5
votes
2answers
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 ...
48
votes
12answers
9k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
4
votes
2answers
611 views

What is the state of research on Horn Angles?

The ancient Greeks struggled with the concept of a horn angle, the "angle" formed by the intersection of two curves. The only information I find in Mathworld is that horn angles are examples of ...
13
votes
1answer
1k views

Fulfilling Pythagoras' Dream using Nonstandard Models of Arithmetic and/or Surreal Numbers

Pythagoras and his followers believed that the Universe was made of numbers. Specifically, they thought that if you compared any magnitudes of the same kind, say the lengths of two objects, you would ...
4
votes
5answers
879 views

“Let $x \in A$”, beginning a proof of “$\forall x \in A$ …”, if A were empty [closed]

I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in ...
71
votes
16answers
16k views

What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here. Had this kind of opinion been expressed before? EDIT. Thanks to all answerers, commentators, voters, ...
13
votes
3answers
1k views

What is the history of the Y-combinator?

Inspired by the comments to this question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus. Where did it first appear? ...
12
votes
3answers
1k views

Has there ever been a weaker Church-like thesis?

Background. The Church-Turing thesis, in one of its many equivalent formulations, states that the intuitively computable arithmetical functions are exactly those computed by Turing machines. ...
29
votes
15answers
5k views

Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...
11
votes
4answers
857 views

Russell and Whitehead's types: ramified and unramified

I was reading Logicomix (a fictionalised account of logic from Frege to Gödel through Russell's eyes) and there was mention about two different versions of types developed by Russell and Whitehead for ...
39
votes
4answers
4k views

What was Gödel's real achievement?

When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but at the fact that the question could be made precise enough to answer: how on earth, even in ...
8
votes
2answers
286 views

Where's the notion of interpretation (model) originally introduced?

Where's the notion of interpretation (model) originally introduced? I find it used in Skolem's paper "Logico-combinatorial investigations in the satisfiability or provability of mathematical ...
6
votes
4answers
1k views

Alternative axiom to induction

Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If ...