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A QUOTE OF WITTGENSTEIN. 'p' is true = p. Ludwig Wittgenstein, Remarks on the foundations of mathematics, Part 1, Appendix 1, Section 6. (This point occurs already in Section 4.442 of the Tractatus Logico-Philosophicus. See the phrase Frege’s "judgement stroke" "⫞" is logically quite meaningless. --- Links to the Tractatus Logico-Philosophicus: https://people.umass.edu/klement/tlp/ --- https://www.wittgensteinproject.org/w/index.php/Project:All_texts )
Here is the whole of Section 6:
6. For what does a proposition's 'being true' mean? 'p' is true = p . (That is the answer.)
So we want to ask something like: under what circumstances do we assert a proposition? Or: how is the assertion of the proposition used in the language-game? And the 'assertion of the proposition' is here contrasted with the utterance of the sentence e.g. as practice in elocution,---or as part of another proposition, and so on.
If, then, we ask in this sense: "Under what circumstances is a proposition asserted in Russell's game?" the answer is: at the end of one of his proofs, or as a 'fundamental law' (Pp.). There is no other way in this system of employing asserted propositions in Russell's symbolism.
Link: https://archive.org/details/remarksonfoundat0000witt_c4e5/page/n7/mode/2up
THE MONTY HALL PROBLEM. Then Monty Hall problem is very famous. A reminder is given below. Here is my preferred solution.
Suppose we have two players, called Switch and No-switch. Switch always switches, No-switch never does. Then Switch wins if and only if she picks a door hiding a goat, which is a probability 2/3 event, whereas No-switch wins if and only if she picks the door hiding the car, which is a probability 1/3 event.
Here is the reminder:
Setting: We have three doors, one door hides a car, each of the other doors hides a goat. Player wins if she reveals the car.
See: Wikipedia entry and A text of Andrew Vazsonyi.
A QUESTION: Do we always have dim End_{End(V)}(V^{**}) = 1 ? Explanation: V is a K-vector space and V^{**} is its second dual. In particular V^{**} is an End(V)-module, and so is the quotient V^{**}/V. As such this quotient has a K-algebra A of endomorphisms. And the question is: Is A always one-dimensional? The answer is trivially yes if V is finite dimensional. More on this: https://mathoverflow.net/q/162162/461
TWO TEXTS
A text about the book Categories and Sheaves by by Masaki Kashiwara and Pierre Schapira --- https://zenodo.org/record/6498631 --- https://vixra.org/abs/1602.0067
A text about the book Introduction to Commutative Algebra by Michael Atiyah and Ian G. MacDonald (including solutions to some exercises) --- https://zenodo.org/record/6498378
OTHER TEXTS
Existence of a least increasing surjection from an artinian poset onto a set of ordinals. We prove that, given an artinian poset X, there is a least increasing surjection f from X onto a set of ordinals. https://github.com/Pierre-Yves-Gaillard/Artin-poset-to-ordinals
Transfinite Recursion --- The only purpose of this short text is to point out that the Transfinite Recursion Theorem is a particular case of the Transfinite Induction Theorem https://github.com/Pierre-Yves-Gaillard/The-Transfinite-Recursion-Theorem
Apparent speed in different inertial frames in special relativity - https://zenodo.org/records/14632096 - In the setting of special relativity, let p be a particle moving at constant velocity. We compare the apparent speeds of p in two inertial frames.
About the Drinker Paradox - https://zenodo.org/records/14599048
Recovering a quadratic form from its orthogonal group --- We prove the following statement: Let K be a field of characteristic different from 2, let V be a K-vector space of finite dimension n > 1, let q and r be two quadratic forms on V, let O(q) and O(r) be the respective orthogonal groups, and let Q, R : V × V → K be the respective associated symmetric bilinear forms. Then we have O(q) = O(r) ⇔ Kq = Kr. https://zenodo.org/records/14513720
Self-dual vector spaces --- Let V be a vector space and G its automorphism group. Say V is *self-dual* if the dual V^* of V is G-isomorphic to V. We answer the question: Which vector spaces are self-dual? https://zenodo.org/records/14361088
A few comments about "Principles of Mathematical Analysis" by Rudin https://zenodo.org/records/13955297
The Jordan-Hölder Theorem https://zenodo.org/records/13870026
Sylow's Theorems https://zenodo.org/records/13854863
Simplicity of A_n https://zenodo.org/records/13844112
Zorns's Lemma. We prove the following statement. Let P be a poset all of whose well-ordered subsets have an upper bound. Then P has a maximal element. https://zenodo.org/records/13791363
The dimension of a finitely generated vector space https://zenodo.org/records/13740597
The Fundamental Theorem of Galois Theory: We give a short and self-contained proof of the Fundamental Theorem of Galois Theory for finite degree extensions. https://vixra.org/abs/1207.0051
Motivating the Gauss sum proof of the quadratic reciprocity. Zenodo. https://doi.org/10.5281/zenodo.6642138
A formula for the Jacobi symbol. Zenodo. https://doi.org/10.5281/zenodo.6721050
The first move in a drawing tic-tac-toe is always optimal - https://doi.org/10.5281/zenodo.10252432
Optimal moves in tic-tac-toe - Zenodo - https://doi.org/10.5281/zenodo.7940866
Natural Ways of Mapping Subsets to Subsets ---10.5281/zenodo.6564983 --- viXra
Finite faithful G-sets are asymptotically free --- 10.5281/zenodo.6573066 --- viXra
Each topological space X is of the form Aut(Y)\Y --- pdf file --- tex file --- (Eric Wofsey found a simpler argument. His argument shows that Y can be required to be T_0. See https://math.stackexchange.com/a/3992100/660)
Any finite connected poset is isomorphic to Aut(X)\X for some finite poset X --- pdf file --- tex file
The Burnside Q-algebras of a monoid --- pdf file --- tex file
The universal profinitization of a topological space --- pdf file --- tex file
WORKS IN PROGRESS
A few comments about ``Topology'' by Munkres https://www.overleaf.com/read/kdwwjvqjrzwb#9fe3a6 - https://github.com/Pierre-Yves-Gaillard/About-Topology-by-Munkres
A few comments about ``Linear Algebra Done Right'' by Axler https://zenodo.org/records/14028516 - https://github.com/Pierre-Yves-Gaillard/About-LADR-by-Axler https://www.overleaf.com/read/fbvxfjsvrgcv#afc0a5