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fan theorem
#1
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Introduction


The fan theorem is one of the basic principles of intuitionism that make it more specific (even in mathematical practice, independent of any philosophical issues) than garden-variety constructive mathematics. In classical mathematics, the fan theorem is true.


Statement


Consider the finite and infinite sequences of binary digits. Given an infinite sequence α and a natural number n, let α¯n be the finite sequence consisting of the first n elements of α.
Let B be a collection of finite sequences of bits, that is a subset of the free monoid on the boolean domain. Given an infinite sequence α and a natural number n, we say that α n-bars B if α¯n∈B; given only α, we say that α bars B if α n-bars B for some n.
We are interested in these three properties of B:
* B is decidable: For every finite sequence u, either u∈B or u∉B. (This is trivial in classical logic but may hold constructively for a particular subset B.)
* B is barred: For every infinite sequence α, α bars B.
* B is uniform: For some natural number M, for every infinite sequence α, if α bars B at all, then α n-bars B for some n≤M.
A bar is a barred subset B.
Fan Theorem. Every decidable bar is uniform.
Although the fan theorem is about bars, it is different from the bar theorem?, which is related but stronger.


Use in analysis


In classical mathematics, the fan theorem is simply true.

In constructive mathematics, the fan theorem is equivalent to any and all of the following statements:

As a locale, Cantor space has enough points (is topological).

As a topological space, Cantor space is compact.

As a topological space, the (located Dedekind) unit interval is compact (the Heine–Borel theorem?).

As a topological space, the (located Dedekind) real line is locally compact.
Every uniformly continuous function from Cantor space to the metric space ℝ˙ + of positive real numbers has a positive lower bound.

Every uniformly continuous function from the unit interval to ℝ˙ + has a positive lower bound.
There exists a class of “kontinuous” partial functions from the set ℝ of (located Dedekind) real numbers to itself (see Waaldijk) such that
o the restriction of a kontinuous function to any smaller domain is kontinuous;
o the identity function on ℝ is kontinuous;
o the composite of two kontinuous functions is kontinuous;
o a function whose domain is the unit interval is kontinuous if and only if it is uniformly continuous (in the usual metric-space sense); and
o the function (x↦1/x) defined on ℝ˙ + is kontinuous.
It follows from any of these statements:
* The bar theorem? holds.
* As a locale, Baire space has enough points.
* Every pointwise-continuous function on Cantor space is uniformly continuous.
* Every pointwise-continuous function on the unit interval is uniformly continuous.
I need to figure out how it relates to the various versions of König's Lemma?, as well as these statements (which are mutually equivalent):
* As a locale, the unit interval has enough points.
* As a locale, the real line (the locale of real numbers) has enough points.
Some of the results above may use countable choice, but probably no more than AC 0,0 (which is choice for relations between ℕ and itself).

منبابع



http://golem.ph.utexas.edu/l
* http://www.jaist.ac.jp/f
* http://www.cairn.info

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#2
سوییچ کردی رو زبان اصلی ا !!
بابا ریاضی فهمیدن فارسی شم سخته چ برسه به زبان اصلی
فکر کنم ویرایش اش کنی بد نباشه، مثل اینکه زبان کیبوردت فارسی بوده
پاسخ
#3
مرسی دقیقا بگو کجاشه؟
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