Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Adding an Abstraction Barrier to ZF Set Theory   Much mathematical writing exists that is, explicitly or implicitly, based on
set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In
ZF, the domain of discourse contains only sets, and hence every mathematical
object must be a set. Consequently, in ZF, with the usual encoding of an
ordered pair ${\langle a, b\rangle}$, formulas like ${\{a\} \in \langle a, b
\rangle}$ have truth values, and operations like ${\mathcal P (\langle a,
b\rangle)}$ have results that are sets. Such 'accidental theorems' do not match
how people think about the mathematics and also cause practical difficulties
when using set theory in machine-assisted theorem proving. In contrast, in a
number of proof assistants, mathematical objects and concepts can be built of
type-theoretic stuff so that many mathematical objects can be, in essence,
terms of an extended typed ${\lambda}$-calculus. However, dilemmas and
frustration arise when formalizing mathematics in type theory.
  Motivated by problems of formalizing mathematics with (1) purely
set-theoretic and (2) type-theoretic approaches, we explore an option with much
of the flexibility of set theory and some of the useful features of type
theory. We present ZFP: a modification of ZF that has ordered pairs as
primitive, non-set objects. ZFP has a more natural and abstract axiomatic
definition of ordered pairs free of any notion of representation. This paper
presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of
the existence of a model for ZFP, which implies that ZFP is consistent if ZF
is. We discuss the approach used to add this abstraction barrier to ZF.

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The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
19
The categories of this paper are: eess
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The categories of this paper are: nlin
2
The categories of this paper are: math
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom