Math Forum / Mathematics / Mathematical Logic / June 2010

I apologize for this fairly ignorant question...but what are the rules
of inference in ZFC?  I read up on the axioms on Wikipedia, but there
was no mention of the rules of inference for proofs in ZFC.  I read
that it is a first-order system of logic, but it didn't specify the

1. You can negate a decision by switching TRUE and FALSE.  For every
recursive set its complement is recursive.
2. You can't negate a set because it is really UNIVERSALSET ^
~SETBEINGNEGATED.  You can take a function of two sets P^~Q but there

As it says, this may be a stupid query.  Be gentle.
1st-order PA has many models.  But it seems to me that there is
a very clear "minimal model", in that the standard model is
(isomorphic to) a subset of any other model.  And I gather

Why are there still any questions about infinity?
We have been able to prove infinity is inconsistent for millennia:
http://en.wikipedia.org/wiki/Zeno's_paradoxes
Even Aristotle couldn't refute Zeno's arguments.

If I were to see a circle on the moon, I would not be surprised.  Nor
if I saw a straight line (line segment.)  But if I saw a square should
I call the newspapers?
If I saw the first 1,000 prime numbers in binary I would definitely

This post refers to Thomas E. Patton, 'Church's Theorem on the Decision
Problem', Notre Dame Journal of Formal Logic, volume VI, Number 2, April
1965.  It will mean nothing to people not familiar with that paper.
Formula (8b) on page 150 reads

http://www.upd.aas.org/had/meetings/2010Abstracts.html
Open Questions Regarding the 1925 Measurement of the Gravitational
Redshift of Sirius B
Jay B. Holberg Univ. of Arizona.

The antidiagonal is too general to meaningfully define a number.
It's not just based on all digits in
forall n, L(n,n)
The antidiagonal argument also has to work on EVERY PERMUTATION of a list.

------------------------SCI.MATH-----------------------------
Take any list of reals
123
456

Is there a way to prove that some relation is r.e. without giving
enough information to make the construction of a program to enumerate
it obvious (extractable from the proof)?
C-B

Consider the Venn Diagram that consists of an overall universal set of
all sets of sentences, and two overlapping sets within it.  One inner
set is the set of r.e. sets, and the other is the set of all co-r.e.
sets (i.e. its complement is r.e.)  Now place these four sets within

There is a space filling medium. It is capable of motion and
resilience. It exerts an expansive pressure in all directions. It is
but a small step from there to the recognition that this very same
material substance is what is formed into the atoms and molecules of

http://www.nature.com/news/2010/100617/full/news.2010.303.html
NATURE: "Gravity is mercilessly impartial - on Earth, it accelerates
light and heavy objects alike with a tug of 9.8 metres per second
squared. That property is the cornerstone of Albert Einstein's theory

It's widely believed our intuition of the natural numbers
has led to foundational understandings of mathematical
reasoning and not the least of which is the validity
of GIT proof. In this post, however, we'll demonstrate

What is the smallest subtree of a given tree that has all of a given
set of nodes within it?
C-B