Re: Peano's Axioms are Inconsistent

On Aug 12, 3:41 pm, Joshua Cranmer <Pidgeo...@xxxxxxxxxxxxxxx> wrote:
RussellE wrote:
On Aug 11, 6:50 pm, Joshua Cranmer <Pidgeo...@xxxxxxxxxxxxxxx> wrote:
And that is the crux of the matter. You can count all the natural
numbers (that is the definition of countable, after all), but at no
point will you have counted all of them.

You can count all the natural numbers and you can't count
all the natural numbers? This sounds like a contradiction to me.

The entire essence of the necessity for distinction is that the size of
the set of natural numbers is not a natural number itself.

Yes, the size of the natural numbers, as defined by Peano's Axioms,
is a natural number. I prove this to be true. The only reason for
"necessity for distinction" is your assumption PA is consistent.

I've already
described why that implies the distinction,

You assume PA is consistent.

but you seem to have been
unable to understand it then and I see no reason why you will be able to
understand it now.

I am feeling the same way.

Including the last 100 blank positions at the end to the tape.

There is no last position, as there is no largest natural number. Your
only evidence for this is an argument using natural (English) language,
which is ambiguous in many cases. Note especially that the concept of
"all" in English predates the understanding of the finer points of
"infinity"  (which itself came around only in the 1800s or thereabouts,
I think)--extrapolation is always a risky adventure, and ambiguous
language even more so.

All doesn't mean all?
I never said BNTM writes to "all" positions.
You do. How are you defining "all"?
"All" means whatever it takes to "prove" PA is consistent?

The old Axiom of Deniability. You simply deny there is
a contradiction. By assuming PA is consistent, you can
"prove" there is no paradox.

Your proof for the contradiction relies entirely on a colloquial usage
of the word "all." Colloquial usages are often different from
mathematical usage; sometimes, some subfields even make distinctions
that others do not. For example, a circle can be considered to be a
sphere in some areas.

At best, you have evidence that letting aleph-null be a natural number
creates a contradiction,

I don't "let" aleph-null be a natural number.
I prove this is true. And I can prove it using
any definition for "all" you care to name,
unless you insist I assume PA is consistent.

but since no one excepts you takes that as an
axiom, it's completely contrary to proving the inconsistency of Peano's
Axioms.

As is any proof, apparently.


Russell
- Integers are an illusion
.

Relevant Pages

  • Re: Peanos Axioms are Inconsistent
    ... This sounds like a contradiction to me. ... I've already described why that implies the distinction, but you seem to have been unable to understand it then and I see no reason why you will be able to understand it now. ... Your proof for the contradiction relies entirely on a colloquial usage of the word "all." ... At best, you have evidence that letting aleph-null be a natural number creates a contradiction, but since no one excepts you takes that as an axiom, it's completely contrary to proving the inconsistency of Peano's Axioms. ...
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