Geodesic
The Cardinal Squaring Principle and an Alternative Axiomatization of NFU
Bulletin of the Section of Logic, Tome 52 (2023) no. 4, pp. 551-581.

Voir la notice de l'article provenant de la source Library of Science

In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about.
Keywords: Quine's New Foundations, cardinal multiplication, axiomatization 
@article{BSL_2023_52_4_a5,
     author = {Adle\v{s}i\'c, Tin and \v{C}a\v{c}i\'c, Vedran},
     title = {The {Cardinal} {Squaring} {Principle} and an {Alternative} {Axiomatization} of {NFU}},
     journal = {Bulletin of the Section of Logic},
     pages = {551--581},
     publisher = {mathdoc},
     volume = {52},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2023_52_4_a5/}
}
TY  - JOUR
AU  - Adlešić, Tin
AU  - Čačić, Vedran
TI  - The Cardinal Squaring Principle and an Alternative Axiomatization of NFU
JO  - Bulletin of the Section of Logic
PY  - 2023
SP  - 551
EP  - 581
VL  - 52
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BSL_2023_52_4_a5/
LA  - en
ID  - BSL_2023_52_4_a5
ER  - 
%0 Journal Article
%A Adlešić, Tin
%A Čačić, Vedran
%T The Cardinal Squaring Principle and an Alternative Axiomatization of NFU
%J Bulletin of the Section of Logic
%D 2023
%P 551-581
%V 52
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BSL_2023_52_4_a5/
%G en
%F BSL_2023_52_4_a5
Adlešić, Tin; Čačić, Vedran. The Cardinal Squaring Principle and an Alternative Axiomatization of NFU. Bulletin of the Section of Logic, Tome 52 (2023) no. 4, pp. 551-581. http://geodesic.mathdoc.fr/item/BSL_2023_52_4_a5/

[1] T. Adlešić, V. Čačić, A Modern Rigorous Approach to Stratification in NF/NFU, Logica Universalis, vol. 16(3) (2022), pp. 451–468 | DOI

[2] H. B. Enderton, Elements of set theory, Academic press (1977) | DOI

[3] H. B. Enderton, A Mathematical Introduction to Logic, Academic Press (2001).

[4] M. J. Gabbay, Consistency of Quine’s New Foundations (2014) | DOI

[5] M. R. Holmes, Systems of combinatory logic related to Quine’s ‘New Foundations’, Annals of Pure and Applied Logic, vol. 53(2) (1991), pp. 103–133 | DOI

[6] M. R. Holmes, The set-theoretical program of Quine succeeded, but nobody noticed, Modern Logic, (1994).

[7] M. R. Holmes, Elementary set theory with a universal set, https://randall-holmes.github.io/head.pdf (1998).

[8] M. R. Holmes, A new pass at the NF consistency proof, https://randall-holmes.github.io/Nfproof/newattempt.pdf (2020).

[9] M. R. Holmes, Proof, Sets, and Logic, https://randall-holmes.github.io/proofsetslogic.pdf (2021).

[10] M. R. Holmes, F. E. Forster, T. Libert, Alternative Set Theories., Sets and extensions in the twentieth century, vol. 6 (2012), pp. 559–632.

[11] R. B. Jensen, On the consistency of a slight (?) modification of Quine’s New Foundations, [in:] J. Hintikka (ed.), Words and objections: Essays on the Work of W. V. Quine, Springer (1969), pp. 278–291.

[12] W. V. Quine, New foundations for mathematical logic, The American mathematical monthly, (1937) | DOI

[13] J. B. Rosser, Logic for mathematicians, Dover Publications (2008) | DOI

[14] G. Wagemakers, New Foundations—A survey of Quine’s set theory, Master’s thesis, Instituut voor Tall, Logica en Informatie Publication Series, X-89-02 (1989).