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Selection Principles in Logic
Kent Mussell
Boise State University
April 21, 2014
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Overview
1
Selection Principles Overview
2
Propositional Logic
3
Selection Principles in Logic
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Selection Principles Overview
Introduction
Selection
Principles
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Selection
Principles
in Logic
Metalogic
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Selection Principles Overview
What Are Selection Principles?
Some examples:
For any sequence of open covers of Q hUn : n ∈ ωi there is a sequence of
single “selections” hsn : n ∈ ωi such that each sn ∈ Un and {sn : n ∈ ω} is
an open cover of Q
For any sequence of open covers of R hUn : n ∈ ωi there is a sequence of
finite “selections” hsn : n ∈ ωi such that each sn ⊆ Un and ∪{sn : n ∈ ω}
is an open cover of R.
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Selection Principles Overview
What Are Selection Principles?
Selection principles are determined by three pieces of information:
1
A family of sets from which any sequence is considered, e.g. the
collection of all open covers of Q or R.
2
A number of “Selections” made from each member of the sequence,
e.g. exactly one selection, finitely many selections, or a number of
selections specified by a function f .
3
A family of sets in which the resulting selections should be contained,
e.g. the collection of all open covers of Q or R.
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Selection Principles Overview
Notation for Common Selection Principles
S1 (A, B):
U1
s1
U2
,
s2
U3
,
s3
...
,
...
Let A and B be families of sets. Then S1 (A, B) denotes the following
statement: For any sequence hUn : n ∈ ωi such that each Un ∈ A there
exists a sequence hsn : n ∈ ωi with each sn ∈ Un such that
{sn : n ∈ ω} ∈ B.
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Selection Principles Overview
Notation for Common Selection Principles
Sfin (A, B):
U1
s1
U2
,
s2
U3
,
s3
...
,
...
Let A and B be families of sets. Then S1 (A, B) denotes the following
statement: For any sequence hUn : n ∈ ωi such that each Un ∈ A there
exists a sequence hsn : n ∈ ωi with each sn ⊆ Un such that
∪{sn : n ∈ ω} ∈ B.
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Propositional Logic
Propositional Logic
Working in propositional logic, we make the following assumptions:
1
The logical language consists of →, ¬, P, Q, R, ... with or without
whole number subscripts. Note: we leave out ∧, ∨, and ↔ because
these can be defined in terms of just → and ¬.
2
Important: Because the lexicon of propositional logic is countable, the
set of all of the formulas of the language is also countable.
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Propositional Logic
Definition of Proof from a set of formulas
Let σ be a set of formulas. A σ-proof is a finite sequence of formulas such
that each member of the sequence is either 1) a member of σ or 2) the
result of applying modus ponens to previous members of the sequence. We
say that a σ-proof is a proof of a formula φ if the last line of the proof is φ.
Definition of Complete
A set of formulas σ is said to be complete if for any formula φ, If φ is a
tautology then there exists a σ-proof of φ.
Definition of Sound
A set of formulas σ is said to be sound if every σ-proof is a proof of a
tautology.
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Propositional Logic
A sound complete set of formulas
The following is a sound and complete set of formulas, Γ. If φ, ψ, and χ
are any well formed formulas then the following are in the set Γ:
1
(φ → (ψ → φ)
2
(φ → (ψ → χ)) → ((φ → ψ) → (φ → χ))
3
(¬ψ → ¬φ) → ((¬φ → φ) → ψ).
For a proof that Γ is both sound and complete see [1].
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Selection Principles in Logic
Selection Principles in Logic
Let A be the set of all sound and complete sets of formulas and let
A = B. We then have:
Claim 1
Sfin (A, B): For any sequence of sound and complete sets of well-formed
formulas, hUn : n ∈ Ni, there is a sequence hsn : n ∈ Ni such that each
sn ⊆ Un and each |sn | is finite and ∪{sn : n ∈ N} is sound and complete.
Claim 2
¬S1 (A, B) : There is a sequence of sound and complete sets of well
formed formulas, hUn : n ∈ Ni such that for any sequence hsn : n ∈ Ni
with |sn | = 1, ∪{sn : n ∈ N} is not both sound and complete.
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Selection Principles in Logic
Selection Principles in Metalogic
The proof of Claim 1 is trivial. There are countably many valid formulas,
so enumerate them φ0 , φ1 , ... For any n we know that Un is complete so
there is a Un -proof of φn , since φn is valid. So select sn to be the members
of Un used to prove φn . This guarantees that ∪{sn : n ∈ N} is sound and
complete.
The proof of Claim 2 is more interesting. We will first need the following
sequence of tautologies:
χ1 = (P → P)
χn+1 = (P → χn )
Claim
Each χn is valid and no χn can be proved from other χn using modus
ponens alone.
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Selection Principles in Logic
Selection Principles in Logic
We can now define the sequence needed in order to prove ¬S1 (A, B). For
each n we define Un as follows:
χn ∈ Un .
If φ ∈ Γ then (χn → φ) ∈ Un .
Nothing else is in Un
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Selection Principles in Logic
What Are the Boundaries of Selection Principles in Logic?
Is there a true selection principle that is logically stronger than Sfin (A, B)
and logically weaker than S1 (A, B)?
The following is false:
Claim 3
There is a function f : N → N such that for any sequence of sound and
complete sets of formulas hUn : n ∈ ωi there is a sequence hsn : n ∈ Ni
such that each sn ⊆ Un , ∪{sn : n ∈ ω} is sound and complete, and
|sn | ≤ f (n).
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Selection Principles in Logic
Proof that Claim 3 is false
Recall the sequence from before χ1 , χ2 , ... Note that it is the countable
(disjoint) union of countable sets. So Let A1 , A2 , A3 , ... be countable sets
such that ∪{An : n ∈ ω} = {χn : n ∈ ω}.
Let f : N → N be an arbitrary function. We can now construct a sequence
of sound and complete sets for which claim 3 fails.
Define a sequence hUn : n ∈ ωi as follows:
For any n, a formula φ is in Un iff φ is of one of the following forms:
1
(A1 → (A2 → ...(Af (n) → ψ))) where ψ ∈ Γ
2
Am where m ≤ f (n)
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Selection Principles in Logic
Further Research
1
Logics other than propositional logic
2
Infinitary Logics
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Selection Principles in Logic
References
[2]Marion Scheepers
Selection principles in topology: New Directions, Filomat (Niš) 15
(2001), 111-126.
[1]Ted Sider (2010)
Logic for Philosophy
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Selection Principles in Logic
The End
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