Finitarily consistent mechanist reasoning and non-finitarily consistent human reasoning: Mutually inconsistent yet complementary!

We now consider the following (tentatively expressed) conclusions suggested by our previous post, which we shall aim to investigate from various perspectives in these pages.

Structures

The Birmingham paper suggests that we may need to distinguish much more sharply than we do at present between:

$\bullet$ Mathematical structures that are built upon only finitary reasoning, and

$\bullet$ Mathematical structures that admit non-finitary reasoning.

Interpretations

For instance the Birmingham paper provides:

$\bullet$ An example of a mathematical structure based on finitary reasoning, namely the finitarily sound algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of the first order Peano Arithmetic PA.

$\bullet$ An example of a mathematical structure based on non-finitary reasoning, namely the non-finitarily sound standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of the first order Peano Arithmetic PA.

Aristotle’s particularisation

The Birmingham paper suggests that the roots of the distinction between these two structures lies in the fact that:

$\bullet$ Finitary reasoning does not assume that Aristotle’s particularisation is always true over infinite domains.

$\bullet$ Non-finitary reasoning assumes that Aristotle’s particularisation is always true over infinite domains.

Consistency of Arithmetic

In the Birmingham paper we also show that:

$\bullet$ Finitary reasoning proves that PA is consistent finitarily (as demanded by the second of Hilbert’s celebrated twenty three problems).

$\bullet$ Non-finitary reasoning proves that PA is consistent non-finitarily (a consequence of Gentzen’s non-finitary proof of consistency for PA).

FOL is consistent; FOL+AP is $\omega$-consistent

This suggests that:

$\bullet$ Finitary reasoning as formalised in first order logic (FOL) is consistent.

$\bullet$ Non-finitary reasoning as formalised in Hilbert’s $\epsilon$-calculus (FOL+AP) is $\omega$-consistent.

$\omega$-consistency

Since the Birmingham paper shows that Aristotle’s particularisation holds over the structure of the natural numbers if, and only if, PA is $\omega$-consistent, it suggests that:

$\bullet$ Finitary reasoning does not admit that PA can be $\omega$-consistent (see Corollary 4 of this post).

$\bullet$ Non-finitary reasoning admits that PA can be $\omega$-consistent.

Arithmetical undecidability

Since proofs of arithmetical undecidability implicitly assume Aristotle’s particularisation, this further suggests that:

$\bullet$ Finitary reasoning does not admit undecidable arithmetical propositions (see Corollary 3 of this post).

$\bullet$ Non-finitary reasoning admits undecidable arithmetical propositions.

Completed Infinity

A significant consequence is that:

$\bullet$ Finitary reasoning does not admit an axiom of infinity.

$\bullet$ Non-finitary reasoning admits an axiom of infinity.

Non-standard models of PA

A further consequence of this is that:

$\bullet$ Finitary reasoning does not admit non-standard models of PA.

$\bullet$ Non-finitary reasoning too does not admit non-standard models of PA.

Algorithmically computable truth and algorithmically verifiable truth

The Birmingham paper also suggests that:

$\bullet$ The truths of finitary reasoning are algorithmically computable.

$\bullet$ The truths of non-finitary reasoning are algorithmically verifiable, but not necessarily algorithmically computable.

Categoricity and incompleteness of Arithmetic

We show in Corollary 1 of this post that it also follows from the Birmingham paper that:

$\bullet$ Finitary reasoning proves that PA is categorical with respect to algorithmically computable truth.

$\bullet$ Non-finitary reasoning proves that PA is incomplete with respect to algorithmically verifiable truth (a consequence of Gödel’s proof of of the undecidability of some arithmetical propositions in any $\omega$-consistent system of arithmetic).

How intelligences reason

This suggests that:

$\bullet$ Finitary reasoning is a shared characteristic of all intelligences, human or non-human.

$\bullet$ Non-finitary reasoning is a characteristic of human intelligence that may not be shared by any other intelligence.

Communication between intelligences: SETI

It further suggests that the search for extra-terrestrial intelligence may benefit from the argument that:

$\bullet$ Finitary reasoning admits effective and unambiguous communication between two intelligences with respect to its (algorithmically computable) arithmetical truths.

$\bullet$ Non-finitary reasoning does not admit effective and unambiguous communication between two intelligences with respect to its (algorithmically verifiable) arithmetical truths.

Determinism, Unpredictability and the EPR paradox

An unexpected consequence of the arguments of the Birmingham paper is that our perspectives on the relation between determinism and predictability may benefit from the paradigm shift demanded by the argument that:

$\bullet$ Finitary reasoning admits the EPR paradox.

$\bullet$ Non-finitary reasoning does not admit the EPR paradox.

The Gödelian argument

The arguments of the Birmingham paper also suggest a fresh perspective on the issue of computationalism since:

$\bullet$ Finitary reasoning does not admit Lucas’ Gödelian argument.

$\bullet$ Non-finitary reasoning admits Lucas’ Gödelian argument.

Effective computability

It further suggests that the nature and status of ‘effective computability’ may also need to be assessed afresh since:

$\bullet$ Finitary reasoning naturally equates algorithmic computability with effective computability.

$\bullet$ Non-finitary reasoning naturally equates algorithmic verifiability with effective computability.

Church Turing Thesis

As also the nature of CT, since:

$\bullet$ Finitary reasoning admits the Church-Turing Thesis.

$\bullet$ Non-finitary reasoning does not admit the Church-Turing Thesis.

Goodstein’s Theorem

Broadly speaking, the two conflicting-but-complementary structures defined in the Birmingham paper suggest that we should be more explicit—in our argumentation—of the structure to which a particular assertion about the natural numbers pertains, since:

$\bullet$ Both finitary and non-finitary reasoning do not admit the proof of Goodstein’s Theorem as neither admits a completed infinity.

$\bullet$ Set-theoretical reasoning admits the proof of Goodstein’s Theorem as it admits a completed infinity.

There’s more …

In the next post we shall consider some further intriguing consequences suggested by the Birmingham paper.

What do you think?

Does Goodstein’s sequence over the natural numbers always terminate or not?