[FOM] Foundational Challenge

[Tennant, Neil]

Reply to Lukasz T. Stepien:

The closure of the usual Dedekind-Peano axioms for arithmetic (assuming they are consistent) is the same in Core Logic as it is in Intuitionistic Logic.

In general: if Intuitionistic Logic provides a proof of conclusion C from a set X of premises, then Core Logic provides a proof either of C or of # (absurdity) from (some subset of) X.

Neil Tennant
________________________________
From: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> on behalf of Lukasz T. Stepien <sfstepie at cyf-kr.edu.pl>
Sent: Monday, January 20, 2020 3:54 AM
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: [FOM] Foundational Challenge


Neil Tennant says:

"Do not blame the strictly classical part of logic for any of the paradoxes.

They all arise for the constructivist.
They can all be derived using only Core Logic (the relevant fragment of Intuitionistic Logic).
And Core Logic suffices for constructive mathematics."


As you know, there in classical logic are paradoxes of material implication related to problem of entailment.

 So, this is interesting issue, whether one can base classical Arithmetic System on Core Logic mentioned by you. My remark is caused by this, what Bernd Buldt has written in his paper: The Scope of Gödel’s First Incompleteness Theorem, Logica Universalis, vol. 8, 499 – 552, (2014), on page 531, namely, that Relevant Arithmetic will be more weak than Classical Arithmetic.

In the paper: T. J. Stepien and L. T. Stepien, "Atomic Entailment and Atomic Inconsistency and Classical Entailment", J. Math. Syst. Sci. 5, No.2, 60-71 (2015)) ; arXiv:1603.06621, a new solution of the problem of entailment has been given - there has been defined atomic entailment and atomic logic has been  formulated. In the next paper: T. J. Stepien and L. T. Stepien, "The Formalization of The Arithmetic System on The Ground of The Atomic Logic", J. Math. Syst. Sci. 5, No.9, 364-368 (2015)) ; arXiv:1603.09334, it has been showed than one can base Classical Arithmetic on atomic logic.

                                                                                                           Lukasz T. Stepien

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