Notes on logical pursuit from the lectures – part one

* Davis-Putnam-Robinson-Matiyasevich work (10H):  the general problem of the existence of integer solutions of Diophantine equations is algorithmically undecidable.
* a proof of consistency of the arithmetical axioms (2H), here Kurt Goedel, negative response from Goedel, euclidean geometry w/o parellel postulate incomplete, use of primitive recursive arithmetic for finitisitc calc, Hilbert-Bernays provability conditions,
* CH, Goedel showed that ZF cannot disprove the CH, Cohen showed that ZF cannot prove the CH, construction of well-ordering of the continuum, FF showed that there exists no definable well-ordering of the continuum,
* Hilbert: we treat mathematics formally as what can be carried out in an axiom system; then we investigate questions of independence, consistency and completeness for these axioms,
* Turing’s result: any true statement is provable at some stage in the transfinite iteration process.
* trade-off: incompleteness for uncertainty about the path due to param space, based on the result from Turing,
* autonomous transfinite progression of axiom systems, are there no genuine absolutely undecidable problems?
* axiom testing: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.
* Feferman: “So, for example, a real number whose square is −1 does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved.”
* logical constants with regards to semantical or inferential criteria
* Lindström: logical only if it is uniformly uniquely characterized by a set of axioms and rules of inference over every universe of discourse.
* Tarski: a unified conceptual apparatus which would supply a common basis for the whole of human knowledge.
* Tarski: paradigmatic use of set theory for the purposes of conceptual analysis.
* Tarski: explication of the notions of truth, of logical consequence and of what is a logical term.
* Tarski: reduces the notion of logical consequence to the notion of truth in a model
* Feferman: “From the structural point of view, our conception is that of a structure (N , 1, Sc, <), where N+ is generated from the initial unit 1 by closure N+ under the successor operation Sc, and for which m < n if m precedes n in the generation procedure. ”
* concepts of continuum: These are: (i) the Euclidean continuum, (ii) Cantor’s continuum, (iii) Dedekind’s continuum, (iv) the Hilbertian continuum, (v) the set of all paths in the full binary tree, and (vi) the set of all subsets of the natural numbers.
* axioms in geometry using betweenness, containment, congruence, based on Hilbert axiom system, with redundant Pasch’s theorem,
* Skolem for set-based approach to logic, multiple incorrect arguments within Penrose Goedelian argument,


About misha

Imagine a story that one can't believe. Hi. Life changes here. Small things only.
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