[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

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.
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.)
>FOM mailing list
>FOM at cs.nyu.edu

Augusto J. Franco de Oliveira
Prof. Emérito Univ. Évora
ajfrancoli at gmail.com

(Este escriba não respeita o AO90.)

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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