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Archive for December, 2010

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Detect a value among others

Hello.

I have a problem, and I don’t know if there is a solution. I couldn’t
find any for the moment. Some help would be nice. Here is the
problem :

Given : a_1, a_2 ,…, a_n, n values and an another one, say a_(n+1).
for each i,j (i!=j) in 1..n a_i != a_j
Aim: Find if the value a_(n+1) is equal to one of the n first. That is
to say there is a k in 1.. such as a_k = a_(n+1)

The fact is I have to do it the simplest way. The idea is to _build_ a
variable with a_1, …, a_n, say f(a_1, …, a_n) and to test for a_(n
+1) with a function g. This will be used in a iterative way.

Finally, one would seek for f and g such as :
g( f(a_1, …, a_n), a_(n+1) ) =
    x if there is a k in 1.. such as a_k = a_(n+1)
    y otherwise (y != x)

I hope this is clear.
Thanks for any help

– Jeremy

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mathematics is not the language of reality

mathematics is not the language of reality
mathematics has no semantic content
mathematics is diviod of semantic content thus it cant say anything about
reality

when meaning is overlayed onto the symbols we end in the Carroll’s
Paradox formalism in mathematics is an attempt to avoid the pitfalls of
Carroll’s Paradox due to semantic meaning being given to the symbols

1+1=2 are just meaningless symbols connected by rules
it is only when we make the symbols correspond to reality that in this
case we see we are dealing with numbers

take the axiomatic system ZFC is just a meaningless array of symbols
connected by rules of inference we give meaning to those symbols and say
ZFC  deals with set

mathematics is diviod of semantic content thus it cant say anything about
reality

as
http://www.mathacademy.com/pr/prime/articles/carroll/index.asp
says

"The formalist solution, while effective, has its own philosophical
drawbacks. Not the least of these is that, by reducing logic to
uninterpreted symbols, all semantic content is removed from the
conclusions of formal logic. In other words, what we would ordinarily
consider meaning is lost. How to restore meaning to systems of inference
while still avoiding difficulties such as Carroll’s Paradox remains a
thorny question for philosophers of mathematics."


Message posted using http://www.talkaboutscience.com/group/sci.logic/
More information at http://www.talkaboutscience.com/faq.html

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Restrictable Primitives?

Hi all,

In a previous discussion, I came to know that if one add a primitive
one place function symbole F to FOL with identity , then the following
becomes a theorem:

Ax Ey ( y=F(x) )

Three questions:-

First: What is the prove of that?

Second: can we adopt a methodology to restrict that? I mean to
restrict F from ranging over all variables of the language, in such a
manner that for a specific formula Phi :

 Ax (Ey(y=F(x))  iff  Phi(x))?

I mean can we have another kind of primive concepts , like a
‘restrictable primitive function symbole’, such that when we say that
F is a restrictable primitive function symbole by Phi , then this
leads to:

Ax (Ey(y=F(x))  iff  Phi(x))

Of course if Phi(x) <-> x=x , it is clear that F becomes unrestricted.

Is that possible?

Third: to generalize matters, if such restrictions are possible, then
can this be generalized over other kinds of primitives?

Zuhair

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Godel's comments about the "true reason" for incompleteness

"The true source of the incompleteness attaching to all formal systems
of mathematics, is to be found—as will be shown in Part II of this
essay—in the fact that the formation of ever higher types can be
continued into the transfinite (c.f., D. Hilbert, ‘Über das
Unendliche’, Math. Ann. 95, p. 184), whereas in every formal system at
most denumerably many types occur. It can be shown, that is, that the
undecidable propositions here presented become always become decidable
by the adjunction of suitable higher types. A similar result also
holds for the axiom system of set theory."

This comment by Godel has me confused, first of all by what he means
by "true source". Isn’t his proof and later refinements/
generalisations of it a "true source" for incompleteness? Also, I was
under the impression that the whole point of Godel’s theorem is that
any kind of proof procedure or list of proof procedures that you can
*even indicate* will not be able to decide all mathematical
propositions. It sounds as if he is saying "ah, we can just continue
adjoining higher types in such and such a manner, and eventually
arbitrary statements become decidable (i.e. for any statement, it
eventually becomes decidable)". I thought the whole point of Godel was
that even if you spent a billion years outlining precisely a method of
coming up with formal systems, there would still be propositions that
could not be resolved by any of those formal systems.

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