Re: Help in answering news story on refutation of fermat's last theorem

Chris Menzel wrote:
> <anzaurres1@xxxxxxxxxxx> said:
> >
> > Yep. I get your point finally. Sorry about it.
>
> Gosh. That's refreshing.

Thank you, Chris. I am dense. Torkel was explaining the foundations of
logic to us at sci.math but I for one was too dense to understand his
extremely cryptic way of putting things. I still am. We have spent
several weeks and dozens of posts in which I was asking Torkel what the
logicians mean by the term "false axiom". And I still fail to
understand the defintions that Torkel had given to us.

Could you logicians please give more detailed explanations of this
probably commonplace term "false axiom"?

Torkel Franzen wrote:
>
> "Mark Nudelman" <markn@xxxxxxxxxxxxxxxxxxxxx> writes:
>
>> I wonder what he thinks it means for an axiom to be "false".
>> Axioms can be inconsistent, but how can they be false?
>
> The same way any other statement can be false. If you take as an
> axiom "0=1", you have a false axiom.

People have asked Torkel for explanations but his answers didn't
explain the common logician concept of a "false axiom" to me.

Torkel Franzen wrote:
> rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin) writes:
>
> > You can't have a ring with only one element?
>
> You can't have an Eiffel Tower that's smaller than a banana? In
> speaking of sentences as true or false, we presuppose some
> interpretation.

Torkel Franzen wrote:
> "Mark Nudelman" <markn@xxxxxxxxxxxxxxxxxxxxx> writes:
>
> If the statement 0=1 is an axiom, then the symbols 0, 1,
>> and = cannot be
>> interpreted as they are in normal arithmetic.
>
> Sure they can. 0=1 simply becomes a false axiom.

Can somebody at sci.logic explain to me at which exact point 0=1
becomes a false axiom? I can understand that "0=1" is a WRONG SATEMENT
about the arithmetic as human beings perceive arithmetic, but what is
the sci.logic definition of a FALSE AXIOM?

Torkel Franzen wrote:
>
>anzaurr...@xxxxxxxxxxx writes:
>
>> I assume by a "false" axiom you mean a statement that is
>> self-contradictory in itself.
>
> Not at all. I simply mean a statement which is false and
> is an axiom of the theory.

> In logic, axiom systems are often studied in order to find
> out what can be proved in them, and there is then nothing
> odd about speaking of false axioms.

I have never heard of a mathematician who would invent some arbitrary
set of axioms just for the pleasure of trying to find out what he can
prove from them, but be as it may, why would one or more axioms of his
invented and arbitrary set of axioms be a "false" axiom? If the set of
axioms has nothing ot do with interpreting reality to begin with, how
can its subset be "false", unless of course this subset is
self-contradictory?

Take the set of 2 axioms to be {"F is a field" and "1 + 1 = 0"}. The
axiom "1 + 1 = 0". is not a false axiom in and of itself, is it? To
me, this axiom would become false only if you put it in some context
like adding an extra axiom like "F is the field of rational numbers".

Torkel Franzen wrote:
> anzaurres1@xxxxxxxxxxx writes:
>
> > You can pick any axiomatic system you want. In some the
> > statement 0=1 will be true. In others - false.
>
> There is no concept of a statement being true or false "in an
> axiomatic system" in logic.

I still don't understand why if my axioms say

1) F is a filed
2) F has more than one element,

why I can't say that it is true that 0 is not equal to 1.

Torkel Franzen wrote:
> anzaurres1@xxxxxxxxxxx writes:
>
> > yes, there exist statements that are false "by itself":
> > A and (not A)
> > where A is prety much any statement of your choice.
>
> This is perfectly arbitrary. I take your statement of the form
> "A and not (A)" to be about bananas.

What does that cryptic response mean in layman's terms?

Torkel Franzen wrote:
> anzaurres1@xxxxxxxxxxx writes:
>
> > > For example, it sometimes prompts them to contradict the simple
> > > observation that there are theories with false axioms.
> >
> > Whom "them"?
>
> It's widespread. You, for example:
>
> > Name one active mathematical
> > non-logician theory, which contains "false axioms", whatever that
> > means.

Where exactly did I contradict the observation that there are theories
with false axioms?

Torkel Franzen wrote:
>
>anzaurres1@xxxxxxxxxxx wrote:
>
>> Torkel Franzen wrote:
>
>>> An odd mathematician, who can't understand explicit
>>> definitions.
>
>> You have an explicit definition? Great. Finally.
>> Please show it to me.
>> If I missed you giving your explicit definition - I apologize in
>> advance.
>
>Yes, a false axiom is an axiom that is false.

I feel like I ask sincere questions and get cryptic mind games in
return.

Can some logician tell me if "a false axiom is an axiom that is false"
is indeed the explicit operational definition that the logicians use?

Are most explicit definitions in contemporary logic similar to this
one? How do logiicans manage to derive new results using such explicit
definitions? I for one wouldn't be able to....

Please, please explain in layman's terms what the concept "false axiom"
means.

Thanks in advance.

.

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