Sets, Truth, and Recursion

Abstract

We discuss some philosophical aspects of an intensional set theory based on an axiomatic truth theory. This set theory gains its justification from natural truth axioms combined with standard recursion-theoretic operations.

The work was partially supported by the ESF research project Dialogical Foundations of Semantics within the ESF Eurocores program LogICCC, LogICCC/0001/2007 and by the projects Hilbert’s Legacy in the Philosophy of Mathematics, PTDC/FIL-FCI/109991/2009, and The notion of mathematical proof, PTDCMHC-FIL/5363/2012, as well as PEst-OE/MAT/UI097/2013, all funded by the Portuguese Science Foundation, FCT.

Appendix

The Applicative Theory TON

${\mathsf{TON}}$ is formulated in ${{\mathcal{L}}_t}$, the first order language of operations and numbers, comprising of individual variables x, y, z, v, w, individual constants ${\mathsf{k}}$, ${\mathsf{s}}$ (combinators), ${\mathsf{p}}$, ${\mathsf{p}_0}$, ${\mathsf{p}_1}$ (pairing and projection), 0, ${\mathsf{s}_\mathsf{N}}$, ${\mathsf{p}_\mathsf{N}}$ (zero, successor and predecessor), ${\mathsf{d}_\mathsf{N}}$ (definition by cases), a binary function symbol ⋅ for term application, and the relation symbols = and $\mathsf{N}$. Terms ($r,s,t,\ldots$) are built up from individual variables and individual constants by term application. Formulas (${\varphi},\psi,\ldots$) are constructed from by ¬, $\mathrel\wedge$ and ${\forall}$ in the usual manner, starting from the atomic formulas $t = s$ and $\mathsf{N}(t)$.

We write $s\,t$ for $(s \cdot t)$ with the convention of association to the left. The connectives ${\vee}$, →, ${\leftrightarrow}$, and ${\exists}$ are defined as usual from the other connectives.

The logic of ${\mathsf{TON}}$ is classical first-order predicate logic with equality, formulated in a Hilbert-style calculus. The non-logical axioms of ${\mathsf{TON}}$ include:

• I Combinatory algebra.

  • (1) ${\mathsf{k}}\,x\,y = x$,

  • (2) ${\mathsf{s}}\,x\,y\,z = x\,z\,(y\,z)$.

• II Pairing and projection.

  • (3) ${\mathsf{p}_0}\,({\mathsf{p}}\,x\,y) = x \mathrel\wedge{\mathsf{p}_1}\,({\mathsf{p}}\,x\,y) = y$.

• III Natural numbers.

  • (4) $\mathsf{N}(0) \mathrel\wedge{\forall} x. \mathsf{N}(x) \to \mathsf{N}({\mathsf{s}_\mathsf{N}}\,x)$,

  • (5) ${\forall} x. \mathsf{N}(x) \to{\mathsf{s}_\mathsf{N}}\,x \not= 0 \mathrel\wedge{\mathsf{p}_\mathsf{N}}\,({\mathsf{s}_\mathsf{N}}\,x) = x$,

  • (6) ${\forall} x. \mathsf{N}(x) \mathrel\wedge x \not= 0 \to \mathsf{N}({\mathsf{p}_\mathsf{N}}\,x) \mathrel\wedge{\mathsf{s}_\mathsf{N}}\,({\mathsf{p}_\mathsf{N}}\,x) = x$.

• IV Definition by cases on $\textsf{N}$.

  • (7) $\mathsf{N}(v) \mathrel\wedge \mathsf{N}(w) \mathrel\wedge v = w \to{\mathsf{d}_\mathsf{N}}\,x\,y\,v\,w = x$,

  • (8) $\mathsf{N}(v) \mathrel\wedge \mathsf{N}(w) \mathrel\wedge v \not= w \to{\mathsf{d}_\mathsf{N}}\,x\,y\,v\,w = y$.

We may add the following natural induction scheme to ${\mathsf{TON}}$:

Formula induction on $\mathsf{N}$ ${\mathop{({{\mathcal{L}}_t}\textsf{-}\mathrm{I}_\textrm{N})}}$

$${\varphi}(0) \mathrel\wedge ({\forall} x. \mathsf{N}(x) \mathrel\wedge{\varphi}(x) \to{\varphi}({\mathsf{s}_\mathsf{N}}\,x)) \to{\forall} x. \mathsf{N}(x) \to{\varphi}(x).$$

${\mathsf{TON}}+{\mathop{({{\mathcal{L}}_t}\textsf{-}\mathrm{I}_\textrm{N})}}$ is proof-theoretically equivalent to Peano arithmetic ${\mathsf{PA}}$.