26
votes
Accepted
Hilbert's sixth problem and QFT description
The reason is that there is no mathematically rigorous construction of any interacting quantum field theory in four space-time dimensions to this date. Because of that, one has not been able so far to ...
- 7,109
10
votes
Accepted
How short can the axioms of propositional logic be?
No. Take the set of truth values to be $\{\top, \bot, P, \neg P, Q, \neg Q\}$, with $\neg$ defined in the obvious way and $\to$ defined by cases: $p \to \top$ and $\bot \to p$ are both $\top$, $\top \...
- 1,411
8
votes
Accepted
Bounded alternatives to powerset that interpret ZFC
The answer is Yes. The simple fact is that it is much easier to interpret ZFC from low-complexity assertions than one might expect. For example, even PA+Con(ZFC) can already interpret ZFC, since one ...
- 225k
7
votes
Accepted
Why not $\sf ZFC+[V=HOD]$?
What does it mean to be a "standard" theory?
By any account, the theory ZFC + V=HOD already is one of the "standard" theories. The axiom V=HOD is intensely studied by set theorists;...
- 225k
6
votes
Accepted
Does Playfair imply Proclus?
I think the following construction gives a counterexample. It stems from the observation that the Playfair axiom is quite weak in the case where all lines only have three points (it produces some ...
- 109k
4
votes
How did Szmielew prove that Pasch's axiom is a consequence of the circle axiom?
No luck here with the source (online issues start at 1973), but I did find the abstract:
The Pasch axiom is known to be independent of the remaining axioms of
the plane Euclidean geometry $E$. By ...
- 178k
4
votes
Accepted
How to use Meredith’s axiom for classical logic?
See https://us.metamath.org/mpeuni/meredith.html and the links there for the proofs you want.
- 39k
2
votes
Axiomatic system made just for playing
Tarski's high school algebra problem involves an axiomatic that has no application as far as I know. It asks whether there are identities involving addition, multiplication, and exponentiation over ...
- 18.5k
2
votes
Harvey Friedman: The expanding mind
Friedman has made public on his website a 2016 draft titled "Expanding Mind Theory". On page 4 there is a definition of a theory $\mathrm{EM}$ in first-order logic which formalizes the ...
- 1,298
1
vote
What are the advantages of the more abstract approaches to nonstandard analysis?
One advantage of the more abstract axiomatic approach to nonstandard analysis was not mentioned when this page was active seven years ago, because the relevant mathematics was not yet available. As ...
- 15.4k
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