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THEOREM 14. For all k ≥ 1, n(k) exists.
BRAIN DEAD: n(1) = 3.
GIFTED HIGH SCHOOL: n(2) = 11.
What about n(3)?? n(3) approx. 60? n(3) approx. 100? n(3)
approx. 200? n(3) approx. 300?
Bad estimates. E.g., n(3) > 2^2^2^2^2^2^2^2^2^2^2^2^2^2^2.
In fact, n(3) > an exponential stack of 2's of length the
A better lower bound: n(3) > the 7198-th level of the
Ackerman hierarchy at 158,386. 12
What about n(4)?? Let A(n) be the n-th level of the Ackerman
hierarchy at n.
THEOREM 15. n(4) > AA...A(1), where there are A(187196) A's.
Now that is a big number.
7. PEANO ARITHMETIC.
Peano arithmetic (PA) is a very fundamental system for f.o.m.
It is the formal system that Gödel used to cast his
Here we give an example of what PA cannot handle. The first
appropriate example of a genuinely combinatorial nature is
Paris/Harrington 1977. Here is a more state of the art
example. Below we use | | for the sup norm.
THEROEM 16. Let n >> k ≥ 1 and F:[0,n]k Æ [0,n]k obey |f(x)|
£ |x|. There exist x1 < ... < xk+1 such that F(x1,...,xk) £
F(x2,...,xk+1) coordinatewise. This cannot be proved in PA.
Predicativity is the view that it is illegitimate to form a
set of integers obeying a property that involves all
sets of integers. One is allowed to form sets of integers
only through definitions that involve sets of integers that
have been previously formed. This view was attrac-tive to
Poincare and Weyl and others.
A modification of this view:
impredicativity is useful for normal mathematics only for the
purpose of proving the existence of infinite sets of integers
of a problematic character.
In this form, the view is demonstrably false, as witnessed
by, say, J.B. Kruskal’s tree theorem 1960:
KRUSKAL’S THEOREM. Let T1,T2,... be finite trees. There exists
i < j such that Ti is continuously embeddable into Tj as
topological spaces. 13
(Continuous embeddability of finite trees is a purely
combinatorial notion, involving only the vertices and the
Kruskal’s proof is blatantly impredicative. Results from
mathematical logic show that under the usual formalizations
of predicativity, there is no predicative proof of Kr...
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