
The Consistency of Arithmetic [pdf] - dstrohmaier
http://timothychow.net/consistent.pdf
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olooney
Odd that this sentence:

> Whereof one cannot speak, thereof one must be silent.

did not get a citation. Perhaps its so well known among its target audience
that none was considered necessary. Since HN appeals to a broader audience,
I'll supply the missing citation: It's from Wittgenstein's Tractatus - the
very last line in fact. The usage is apt - as far as I can tell, Wittgenstein
was making the same point about the difficult - even impossibility - of making
meaningful statements outside of a formal language.

[https://en.wikipedia.org/wiki/Tractatus_Logico-
Philosophicus...](https://en.wikipedia.org/wiki/Tractatus_Logico-
Philosophicus#Proposition_7)

~~~
Quekid5
Re: making meaningful statements outside of a formal language.

Well, yes I guess that makes sense if we're talking "meaningful" in a formal
sense... but then we're again stuck with the definition of what "meaningful"
is.

If I don't agree with your definition of "meaningful", then why would I listen
to "your" philosophy? (EDIT: This is not meant to be personally
confrontational, just a blunt statement of the essential problem.)

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Isamu
Very readable, and fun to read. More please!

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skybrian
This is great!

I'd love to read a similar explanation about surreal numbers. Are surreal
numbers in N or not? How do we know? (In my limited understanding, they're not
easily excluded.)

~~~
Sniffnoy
The surreals have no relation to any of this. I'm guessing you've confused
them with nonstandard natural numbers? In which case the answer is, well, what
are we assuming? From the point of view of ZFC, of course there are no
nonstandard naturals -- but PA can't prove this. Note that this is from the
point of ZFC that PA can't prove this, since PA itself can't even formalize
the notion of a nonstandard natural number (if it could, it could prove they
don't exist).

In any case, surreal numbers are an entirely different system of numbers that
exist in ordinary mathematics. As opposed to nonstandard naturals, which are a
"what if we look at other ways of doing math?" thing.

~~~
skybrian
Yes, nonstandard natural numbers are what I meant. Sorry!

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jesuslop
In Gödel's collected works vol. 2 one has the detailed argument by transfinite
induction up to the first inaccesible ordinal. Interesting to have this as
guidemap.

~~~
ASipos
By "first inaccessible ordinal" you mean epsilon_0? (Probably you mean it in
the sense of the first one inaccessible by addition, multiplication and
exponentiation.) Because there is also a "first recursively inaccessible
ordinal", which is much larger (also, not recursive: it's used to construct,
by collapsing, the ordinal of Delta12-CA+BI, which is also much larger though
recursive).

~~~
jesuslop
Yes I was being sloppy. epsilon_0 as the limit of iterating ordinal
exponentiation by omega. Also the correct volumen is number 3.

