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The Independence of Peano's Fourth Axiom from MartinLöf's Type Theory without Universes
 Journal of Symbolic Logic
, 1988
"... this paper will work for any of the different formulations of MartinLof's type theory. 2 The construction of the interpretation ..."
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this paper will work for any of the different formulations of MartinLof's type theory. 2 The construction of the interpretation
The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
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SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
INTERNAL AND EXTERNAL CONSISTENCY OF ARITHMETIC
"... What Gödel referred to as “outer ” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of ari ..."
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What Gödel referred to as “outer ” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation ” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent ” (Modern Logic, vol. 8, n ° 12, 2000, pp. 4786). An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s
The Failure of Leibniz’s Infinite Analysis view of Contingency
"... Abstract: In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’ ..."
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Abstract: In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’s "infinite analysis " view of contingency is incorrect. It is then argued that Leibniz's own examples of incommensurable lines and convergent series undermine, rather than bolster his view by providing examples of necessary mathematical truths that are not demonstrable. Finally, it is argued that a more modern view on convergent series would, in certain respects, help support some claims he makes about the necessity of mathematical truths, but would still not yield a viable theory of necessity due to remaining problems with other logical, mathematical, and modal claims. From his early metaphysical writings, such as “On Freedom and Possibility ” to his later writings such as “The Monadology”, Leibniz distinguished between those propositions which are necessary and those that are contingent. A central problem for Leibniz is to explain how there could be such a distinction. Since all truths necessarily follow from God’s choice to actualize this world, (truths that Leibniz calls hypothetically or morally necessary), it seems that since God