Corpus evidence — top 10 passages

Most-relevant passages from the entire indexed corpus (67,286 paragraph chunks across YouTube transcripts, PubMed, arXiv, archive.org, Stanford Encyclopedia of Philosophy, OpenAlex, and more) ranked by semantic similarity (bge-small-en-v1.5).

1. 01 · blog0.774

   When \(\vdash\) has been described by means of some proof system (some collection of rules), then the “only if” (“if”) direction of this claim amounts to the claim that \(\vdash\) is sound (respectively, complete ) with respect to \(V\). Take the language whose sole primitive connective is \(\wedge\) and the class of \(\wedge\)-Boolean valuations. A simple proof system is provided by the rules indicated in the obvious way by the following figures: \[ (\wedge\text{I})\frac{\phi\quad\psi}{\phi\wedge\psi} \qquad (\wedge\text{E})\frac{\phi\wedge\psi}{\phi}, \frac{\phi\wedge\psi}{\psi} \] The rule …

   blog/plato-stanford-edu/sentence-connectives-in-formal-logic.md

2. 02 · blog0.773

   We are asking how many natural truth theoretic axioms one can add to CT - [PA], assuring this way that the resulting theory does not allow for pathologies of a certain kind, so that the theory remain on the conservative side of the boundary. Since Tarski was the first to obtain results about the deductive weakness of some axiomatic theories of truth, the boundary was called the Tarski Boundary (name suggested by Ali Enayat). One approach to the problem of delineating the contour of the Tarski Boundary is to study the strength of subsystems of CT(PA) with induction restricted to certain class o…

   blog/plato-stanford-edu/axiomatic-theories-of-truth.md

3. 03 · blog0.768

   For the \(\vdash_{\wedge}\)-specific point, see the discussion of Gabbay’s result on the projection-conjunction truth-functions below on the precise extent of the feature in question, which, for example, none of \(\vee , \rightarrow , \neg\) exhibits.) The significance of this is that if one started not with the class of \(\wedge\)-Boolean valuations (or any other kind of valuational semantics), but with the rules \((\wedge\text{I})\) and \((\wedge\text{E})\) themselves, then that class of valuations would force itself on one’s attention anyway as the only class of valuations, preservation of …

   blog/plato-stanford-edu/sentence-connectives-in-formal-logic.md

4. 04 · yt0.768

   So now the topology is   all messed up. These are going to have the  same causal structure because every point   is causally related to every other in both of  them. So there's a causal isomorphism. They   have the same causal structure, but  they have very different topologies. And the beauty of what David did is he said,  how weak can we go here in terms of what's   the minimal level of causal structure that we  need to show that causal structure determines   the shape of the universe? And so he went  to work on …

   yt/iGOGxaZZHwE-it-s-not-that-we-don-t-know-it-s-that-we-can-t/transcript.txt

5. 05 · blog0.765

   The generalized quantifier axioms, which expresses that for every \(z\) the result of prefixing a formula \(\phi(\bar{x})\) with a string of quntifiers \(\forall x_z\ldots\forall x_1\) is true if and only if every formula resulsting for substituting numerals for variables \(x_z,\ldots,x_1\)in \(\phi(\bar{x})\) is true: \(\forall \phi(\bar{x})\forall z\bigl(T[\forall x_z\ldots\forall x_1\phi(\bar{x})]\leftrightarrow \forall \alpha \bigl(\text{Asn}(\alpha,z)\rightarrow T[\phi[\alpha]]\bigr).\) The regularity principle, expressing that every two terms which denote the same number are intersubstit…

   blog/plato-stanford-edu/axiomatic-theories-of-truth.md

6. 06 · blog0.765

   \] Since evidently we may assert \(\Phi(U)\) and \(\Phi(V)\), it follows from (2) that we may assert \(U(KU)\) and \(V(KV)\), whence also, using (1), \[ [A \vee KU = 0] \wedge [A \vee KV = 1]. \] Using the distributive law (which holds in intuitionistic logic), it follows that we may assert \[ A \vee [KU = 0 \wedge KV = 1]. \] From the presupposition that \(0 \ne 1\) it follows that \[ \tag{3} A \vee KU \ne KV \] is assertable. But it follows from (1) that we may assert \(A \rightarrow U ≈ V\), and so also, using the Extensionality of Functions, \(A \rightarrow KU = KV\). This yields the asser…

   blog/plato-stanford-edu/the-axiom-of-choice.md

7. 07 · yt0.764

   You you set these are the rules that we are we adopt to prove things. Then you what Gödel shows is an amazing thing. I always thought it was amazing. There is a statement which you by virtue of your trust in these rules, you can see that it's true. Yet, you can't prove it by the rules. Now, I found this absolutely amazing because it means you're you don't use the rules to to understand things because how do you know this thing is true? Well, you know it's true but because you trust the rules. Well, it's you're you're if you're using the rules, then how do you know that using the rules only giv…

   yt/OoDi856wLPM-sir-roger-penrose-stuart-hameroff-collapsing-a-theory-of-qua/transcript.txt

8. 08 · blog0.764

   1963 Paul Cohen proves independence of AC from the standard axioms of set theory (Cohen 1963, 1964). 2. Independence and Consistency of the Axiom of Choice As stated above, in 1922 Fraenkel proved the independence of AC from a system of set theory containing “atoms”. Here by an atom is meant a pure individual, that is, an entity having no members and yet distinct from the empty set (so a fortiori an atom cannot be a set). In a system of set theory with atoms it is assumed that one is given an infinite set \(A\) of atoms. One can build a universe \(V(A)\) of sets over \(A\) by starting with \(A…

   blog/plato-stanford-edu/the-axiom-of-choice.md

9. 09 · yt0.763

   I can only see the red and green part but the is controlled by the bigger matrix. Right? So that little matrix that I get just on red and green is called a trace. So that so that's called the trace. So for any big matrix I can look at any subset of its states like take red, green, blue and yellow. I can just trace onto red green or blue and yellow or red and yellow. There's all all sorts of combin I can take a trace on any subset that I want to. So very very you know simple idea but theiite the nice thing is it's it's in general unique. There's you give me a big matrix and you tell me which st…

   yt/Hf1q-bZMEo4-what-are-traces-of-consciousness-a-new-breakthrough-unifying/transcript.txt

10. 10 · blog0.763

    The “proofs” are certain robust sets of reals (universally Baire sets of reals) and the test structures are models that are “closed” under these proofs. The precise notions of “closure” and “proof” are somewhat technical and so we will pass over them in silence. [ 7 ] Like the semantic relation, this quasi-syntactic proof relation is robust under large cardinal assumptions: Theorem 3.6 (Woodin 1999). Assume ZFC and that there is a proper class of Woodin cardinals. Suppose T is a countable theory in the language of set theory, φ is a sentence, and B is a complete Boolean algebra. Then T ⊢ Ω φ i…

    blog/plato-stanford-edu/the-continuum-hypothesis.md