Articles of meta math

Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, “Does Mathematics Need New Axioms?” The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. 401-446 John Steel makes the following remark in footnote 29: There is the very remote possibility that one could show […]

Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted comprehension: Given any predicate $\varphi$ in the language of set theory with one free variable $x$ (and perhaps some […]

What are some natural arithmetical statements independent of ZFC?

Gödel’s first incompleteness theorem produces a statement in the language of arithmetic that’s independent of a given theory. The second theorem says that a consistient theory can not prove its own consistency, which is also a arithmetical statement (since you phrase it in terms of a Turing machine that looks for contradictions, for example). Are […]

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) 😉 The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe – in many cases, as Gowers points out in his amazing book “Mathematics. A very short intro”, it is mainly about approximations, […]

The standard role of intuitive numbers in the foundations of mathematics

In my career I’ve been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about the common “rules” that are accepted in meta-math and, more specifically, in the foundations of mathematics. I have several questions, […]

Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some other reasonable explanation for why we have only identified interesting small constants? EDIT: I may have […]

Why is it considered unlikely that there could be a contradiction in ZF/ZFC?

EDIT: No answer addresses the “bottleneck” question. It’s not surprising to me because the question is vague. But I would like to know whether that is indeed the reason, or perhaps something else. The question is interesting to me and I would be grateful for any help with it. MAIN PART: Unfortunately, there will be […]

How do you go about doing mathematics on a day to day basis?

Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ideas. A venerable mathematician once described his career to me as constantly stumbling around in the dark. Of course, this […]

Meaning of symbols $\vdash$ and $ \models$

I’m confused about the use of symbols $\vdash$ and $ \models$. Reading the answers to Notation Question: What does $\vdash$ mean in logic? and What is the meaning of the double turnstile symbol ($\models$)? I see that: the turnstile symbol $ \vdash $ denotes syntactic implication. Then $ S \vdash \psi$ means that $\psi$ can […]

Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?

OK, maybe the title is exaggerated, but is it true that the rest of math is just “good enough”, or a good approximation of absolute truth – like Newtonian physics compared to general relativity? How do we know that our “approximation” is the right one? Another analogy: Fundamental physics is also not well-fundamented (where is […]