[FOM] "Proof" of the consistency of PA published by Oxford UP
A J Franco de Oliveira
francoli at kqnet.pt
Sat Mar 7 07:40:34 EST 2015
Hello
This is a response to you only.
How could one define "potential infinite"? I
believe that there is a possible approximation of
a "potential infinite" axiom in E. Nelson´s
Internal Set Theory (1977): Internal set theory:
A new approach to nonstandard analysis. Bulletin
of the American Mathematical Society 83(6):1165–1198.
In this conservative extension of ZFC (a similar
extension of PA exists) the definable collection
of standard natural numbers is not a
set (although it is said to be an external set)
and so is not an infinite set but is a good
approximation to a potential infinite set as one
can get: the number 0 is standard and for every
standard natural number n, the sucessor of n is
standard. The said collection also satisfies an
"external induction principle". In every infinite
set we can obtain a copy of the standard natural
numbers which is not a set. So what I propose is
that the idea of "potential infinite" can be
realized by any copy of the standard natural numbers.
Regards
ajfo
At 23:11 06-03-2015, you wrote:
>Richard Heck wrote:
>
>>If there's something interesting here, it's the
>>way the semantics he develops doesn't require
>>there to be a single infinite model, but only a
>>succession of every-larger finite models. There
>>are antecedents to that sort of idea in modal
>>structuralist views, I believe, of the sort
>>developed by Hellman, and perhaps more than
>>antecedents. Maybe there are more developed
>>forms of this idea, too, and if so I'd be interested to know where.
>
>There is of course a long tradition in
>philosophy of distinguishing between "potential
>infinity" and "actual infinity." In modern
>mathematics, this distinction doesn't seem to
>exist. The closest thing to "potential
>infinity" seems to be an axiomatic system that
>lacks anything that could be identified as an
>explicit "axiom of infinity," yet admits only
>(actually) infinite models. (PA would be an
>example.) But for example, I've never seen
>anyone define two separate axioms and declare
>one of them to be an "axiom of potential
>infinity" and the other an "axiom of actual infinity."
>
>I'm wondering if anyone can come up with (or has
>already come up with) candidates for two such
>axioms, the former demonstrably weaker than the
>latter, such that the consistency of PA can be
>proved using only the "axiom of potential
>infinity." Doing this might be pleasing to
>those who not only perceive an important
>distinction between potential and actual
>infinity but go so far as to reject the latter
>while accepting the former. (McCall seems to be
>one such person, if I read him correctly.)
>
>Tim
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>
Augusto J. Franco de Oliveira
Prof. Emérito Univ. Évora
CFCUL
ajfrancoli at gmail.com
(Este escriba não respeita o AO90.)
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http://cfcul.fc.ul.pt/equipa/3_cfcul_elegiveis/franco_oliveira/afrancoliveira.htm
in http://cfcul.fc.ul.pt/
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O número dos tolos e dos cegos continua sendo
infinito, como nos tempos bíblicos. Título da
Parte V do livro de Tomás da Fonseca (1887-1968),
NA COVA DOS LEÕES, Lisboa, 955 (edição
fac-simile com o título FÁTIMA, A Bela e o Monstro, Lisboa, 2014)
Continuo à procura dela (da esperança) para
perceber onde é que os meus filhos e netos irão viver.
A grande desvantagem de ser velho é perceber que
pouco ou nada muda. John Le Carré, in Suplemento
"Actual" do EXPRESSO de 27-04-2013.
You can only find truth with logic if you have
already found truth without it. Gilbert Keith Chesterton (1874-1936)
Quem sabe, faz, quem compreende, ensina. Aristóteles
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