超实数系统存在性的严格数学证明
2019-08-13 06:31阅读：
超实数系统存在性的严格数学证明
1929年，23岁“小毛头”哥德尔证明了“完全性定理”（if…
then..）如下：
“The completeness theorem says that if a formula
is logically valid then there is a finite deduction (a formal
proof) of the
formula”.（后来，到了1953年，该定理的证明过程被简化）。
基本思路是，由该定理出发，导出紧致性定理，而紧致行定理的一个简单推论就是超实数系统存在性的数学证明（由鲁宾逊完成）。
什么是紧致性定理？请见本文附件。
面对如此情景。莱布尼兹望而生叹。
袁萌 陈启清
8月12日
附件?
Compactness theorem
In mathematical logic, the compactness theorem states that a
set of
first-order sentences has a model if and only if every finite
subset of it has a model. This theorem is an important tool in
model theory, as it provides a useful method for constructing
models of any set of sentences that is finitely consistent.
The compactness theorem for the propositional calculus is a
consequence of Tychonoff's theorem (which says that the product of
compact spaces is compact) applied to compact Stone spaces[1],
hence the theorem's name. Likewise, it is analogous to the finite
intersection property characterization of compactness in
topological spaces: a collection of closed sets in a compact space
has a non-empty intersection if every finite subcollection has a
non-empty intersection.
The compactness theorem is one of the two key properties,
along with the downward Löwenheim–Skolem theorem, that is used in
Lindström's theorem to characterize first-order logic. Although
there are some generalizations of the compactness theorem to
non-first-order logics, the compactness theorem itself does not
hold in them.
Contents
1 History
2 Applications
3 Proofs
4 See also
5 Notes
6 References
History
Kurt Gödel proved the countable compactness theorem in 1930.
Anatoly Maltsev proved the uncountable case in
1936.[2][3]
Applications
The compactness theorem has many applications in model
theory; a few typical results are sketched here.
The compactness theorem implies Robinson's principle: If a
first-order sentence holds in every field of characteristic zero,
then there exists a constant p such that the sentence holds for
every field of characteristic larger than p. This can be seen as
follows: suppose φ is a sentence that
holds in every field of characteristic zero. Then its negation
¬φ, together with the field axioms and
the infinite sequence of sentences 1+1 ≠
0, 1+1+1 ≠ 0,
…, is not satisfiable (because there is no field
of characteristic 0 in which ¬φ holds, and the infinite sequence of
sentences ensures any model would be a field of characteristic 0).
Therefore, there is a finite subset A of these sentences that is
not satisfiable. We can assume that A contains ¬φ, the field
axioms, and, for some k, the first k sentences of the form
1+1+...+1 ≠ 0 (because adding more
sentences doesn't change unsatisfiability). Let B contain all the
sentences of A except ¬φ. Then any field
with a characteristic greater than k is a model of B, and
¬φ together with B is not satisfiable.
This means that φ must hold in every model of B, which means
precisely that φ holds in every field of characteristic greater
than k.
A second application of the compactness theorem shows that
any theory that has arbitrarily large finite models, or a single
infinite model, has models of arbitrary large cardinality (this is
the Upward Löwenheim–Skolem theorem). So, for instance, there are
nonstandard models of Peano arithmetic with uncountably many
'natural numbers'. To achieve this, let T be the initial theory and
let κ be any cardinal number. Add to the language of T one constant
symbol for every element of κ. Then add to T a collection of
sentences that say that the objects denoted by any two distinct
constant symbols from the new collection are distinct (this is a
collection of κ2 sentences). Since every finite subset of this new
theory is satisfiable by a sufficiently large finite model of T, or
by any infinite model, the entire extended theory is satisfiable.
But any model of the extended theory has cardinality at least
κ
A third application of the compactness theorem is the
construction of nonstandard models of the real numbers, that is,
consistent extensions of the theory of the real numbers that
contain 'infinitesimal' numbers. To see this, let Σ be a
first-order axiomatization of the theory of the real numbers.
Consider the theory obtained by adding a new constant symbol ε to
the language and adjoining to Σ the axiom ε > 0 and the axioms ε
< 1/n for all positive integers n. Clearly, the standard real
numbers R are a model for every finite subset of these axioms,
because the real numbers satisfy everything in Σ and, by suitable
choice of ε, can be made to satisfy any finite subset of the axioms
about ε. By the compactness theorem, there is a model *R that
satisfies Σ and also contains an infinitesimal element ε. A similar
argument, adjoining axioms ω > 0, ω > 1, etc., shows that the
existence of infinitely large integers cannot be ruled out by any
axiomatization Σ of the reals.[4]
Proofs
One can prove the compactness theorem using Gödel's
completeness theorem, which establishes that a set of sentences is
satisfiable if and only if no contradiction can be proven from it.
Since proofs are always finite and therefore involve only finitely
many of the given sentences, the compactness theorem follows. In
fact, the compactness theorem is equivalent to Gödel's completeness
theorem, and both are equivalent to the Boolean prime ideal
theorem, a weak form of the axiom of choice.[5]
Gödel originally proved the compactness theorem in just this
way, but later some 'purely semantic' proofs of the compactness
theorem were found, i.e., proofs that refer to truth but not to
provability. One of those proofs relies on ultraproducts hinging on
the axiom of choice as follows:
Proof: Fix a first-order language L, and let Σ be a
collection of L-sentences such that every finite subcollection of
L-sentences, i ⊆ Σ of it has a
model
M i {\displaystyle {\mathcal {M}}_{i}}
. Also let
∏ i ⊆ Σ M
i {\displaystyle \prod _{i\subseteq \Sigma }{\mathcal
{M}}_{i}}
be the direct product of the structures
and I be the collection of finite subsets of Σ. For each i in I let
Ai := { j ∈ I : j ⊇
i}. The family of all of these sets Ai generates a proper
filter, so there is an ultrafilter U containing all
sets of the form Ai.
Now for any formula φ in Σ we have:
the set A{φ} is in U
whenever j ∈
A{φ}, then φ
∈ j, hence φ holds
in
M j {\displaystyle {\mathcal {M}}_{j}}
the set of all j with the property that φ holds
in
M j {\displaystyle {\mathcal {M}}_{j}}
is a superset of A{φ}, hence also in
U
Using o's theorem we see that φ holds in the
ultraproduct
∏ i ⊆ Σ M i / U
{\displaystyle \prod _{i\subseteq \Sigma }{\mathcal
{M}}_{i}/U}
. So this ultraproduct satisfies all formulas in
Σ.
See also
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