After the paradoxes
Historical introduction to the philosophy of mathematics
András Máté
21th October 2016
András Máté
After the paradoxes
Aims of foundational research
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
Create rich (possibly omniscient) theories
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
Create rich (possibly omniscient) theories
Three ways out:
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
Create rich (possibly omniscient) theories
Three ways out:
1
Improve logic and produce a unique general theory (logicism)
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
Create rich (possibly omniscient) theories
Three ways out:
1
Improve logic and produce a unique general theory (logicism)
2
Risky theories but a reliable metatheory (formalism)
András Máté
After the paradoxes
Aims of foundational research
Create firmly–based/indubitable theories
Avoid inconsistency
Create rich (possibly omniscient) theories
Three ways out:
1
Improve logic and produce a unique general theory (logicism)
2
Risky theories but a reliable metatheory (formalism)
3
Abandon the priority of logic in favor of a more reliable basis
(intuitionism)
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After the paradoxes
Hilbert’s foundational program at the beginnings
András Máté
After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
András Máté
After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
Improved version of Euclid
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After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
Improved version of Euclid
Change of perspective: not the science of the space, but the
science of anything that satisfies the axioms.
András Máté
After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
Improved version of Euclid
Change of perspective: not the science of the space, but the
science of anything that satisfies the axioms.
Another programmatic claim: existence in mathematics is nothing
but consistency
András Máté
After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
Improved version of Euclid
Change of perspective: not the science of the space, but the
science of anything that satisfies the axioms.
Another programmatic claim: existence in mathematics is nothing
but consistency
Ideal case: categorical theories
András Máté
After the paradoxes
Hilbert’s foundational program at the beginnings
Hilbert (Foundations of Geometry, 1st edition 1899): Axioms
implicitly definite the basic notions of the theory.
Improved version of Euclid
Change of perspective: not the science of the space, but the
science of anything that satisfies the axioms.
Another programmatic claim: existence in mathematics is nothing
but consistency
Ideal case: categorical theories
Standard way to investigate theories in the metatheory: formalize
them in first-order logic.
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After the paradoxes
First-order logic
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After the paradoxes
First-order logic
Let us have a sequence of closed sentences Γ in some L (1)
first-order language.
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After the paradoxes
First-order logic
Let us have a sequence of closed sentences Γ in some L (1)
first-order language.
Thm(Γ ) is the set of sentences derivable from Γ .
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After the paradoxes
First-order logic
Let us have a sequence of closed sentences Γ in some L (1)
first-order language.
Thm(Γ ) is the set of sentences derivable from Γ .
Γ is consistent iff Thm(Γ ) is a proper subset of the sentences of
L (1) .
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After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
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After the paradoxes
0
(1)
that is an
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
it contains every member of Γ ;
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After the paradoxes
0
(1)
that is an
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
if it contains ¬(A → B), then it contains both A and ¬B;
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
if it contains ¬(A → B), then it contains both A and ¬B;
if it contains ¬∀xA, then it contains at least one sentence of
the form ¬A(a/x) ;
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
if it contains ¬(A → B), then it contains both A and ¬B;
if it contains ¬∀xA, then it contains at least one sentence of
the form ¬A(a/x) ;
if it contains ∀xA, then it contains
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
if it contains ¬(A → B), then it contains both A and ¬B;
if it contains ¬∀xA, then it contains at least one sentence of
the form ¬A(a/x) ;
if it contains ∀xA, then it contains
at least one sentence of the form A(a/x) ;
András Máté
After the paradoxes
Analytic sequences
The sequence of closed sentences Γ ∗ (in some L
extension of L (1) by some in-constants) is a
finished analytic sequence to Γ iff
0
(1)
that is an
it contains every member of Γ ;
if it contains a sentence ¬¬A, then it contains A, too;
if it contains A → B, then it contains either ¬A or B;
if it contains ¬(A → B), then it contains both A and ¬B;
if it contains ¬∀xA, then it contains at least one sentence of
the form ¬A(a/x) ;
if it contains ∀xA, then it contains
at least one sentence of the form A(a/x) ;
every sentence of the form A(a/x) where a is any in-constant
occurring in Γ ∗ ;
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After the paradoxes
Analytic sequences (continuation)
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
András Máté
After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
András Máté
After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
every step of the algorithm produces a consistent extension of Γ
and either
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
every step of the algorithm produces a consistent extension of Γ
and either
I Γ ∗ is a finished analytic sequence to Γ or
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After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
every step of the algorithm produces a consistent extension of Γ
and either
I Γ ∗ is a finished analytic sequence to Γ or
II Γ ∗ is finite and contains a contradiction.
András Máté
After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
every step of the algorithm produces a consistent extension of Γ
and either
I Γ ∗ is a finished analytic sequence to Γ or
II Γ ∗ is finite and contains a contradiction.
In case I, Γ ∗ has a model (therefore Γ has a model, too) whose
domain is a subset of NN.
András Máté
After the paradoxes
Analytic sequences (continuation)
if it contains an atomic sentence A and a sentence a = b, then
it contains both A(a/b) and A(b/a)
it contains no contradiction, i.e.
no sentence of the form a 6= a;
no pair of sentences A, ¬A.
Proposition There is an algorithm that produces a sequence Γ ∗ of
closed sentences from Γ s.t.
every step of the algorithm produces a consistent extension of Γ
and either
I Γ ∗ is a finished analytic sequence to Γ or
II Γ ∗ is finite and contains a contradiction.
In case I, Γ ∗ has a model (therefore Γ has a model, too) whose
domain is a subset of NN.
In case II, Γ is inconsistent.
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Theorems
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
Or again, if a sentence is derivable from a set of sentences, then it
is derivable from a finite subset of it.
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
Or again, if a sentence is derivable from a set of sentences, then it
is derivable from a finite subset of it.
Completeness: Every consistent set of sentences has a model.
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
Or again, if a sentence is derivable from a set of sentences, then it
is derivable from a finite subset of it.
Completeness: Every consistent set of sentences has a model.
If every finite subset of Γ has a model, then Γ has a model, too.
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After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
Or again, if a sentence is derivable from a set of sentences, then it
is derivable from a finite subset of it.
Completeness: Every consistent set of sentences has a model.
If every finite subset of Γ has a model, then Γ has a model, too.
If a sentence C is a (semantical) consequence of a set of sentences
Π (i.e., Π ∪ {¬C} has no model), then C is derivable from Π.
András Máté
After the paradoxes
Theorems
Compactness: If Γ is inconsistent, then it has an inconsistent finite
subset.
In other words, if every finite subset of Γ is consistent, then Γ is
consistent, too.
Or again, if a sentence is derivable from a set of sentences, then it
is derivable from a finite subset of it.
Completeness: Every consistent set of sentences has a model.
If every finite subset of Γ has a model, then Γ has a model, too.
If a sentence C is a (semantical) consequence of a set of sentences
Π (i.e., Π ∪ {¬C} has no model), then C is derivable from Π.
Löwenheim-Skolem: If a set of sentences has a model, then it has
a countable model, too.
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After the paradoxes
Some consequences
Let us suppose that we have a first-order theory of real numbers
that contains the usual operations and relations for real numbers
and proves at least some simple propositions about them. Let us
suppos e further than we have a model of it that consists of the
real numbers as we are thinking about them ("standard model").
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After the paradoxes
Some consequences
Let us suppose that we have a first-order theory of real numbers
that contains the usual operations and relations for real numbers
and proves at least some simple propositions about them. Let us
suppos e further than we have a model of it that consists of the
real numbers as we are thinking about them ("standard model").
We can prove within the theory that the domain (the set of real
numbers) is not countable (Cantor’s theorem).
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After the paradoxes
Some consequences
Let us suppose that we have a first-order theory of real numbers
that contains the usual operations and relations for real numbers
and proves at least some simple propositions about them. Let us
suppos e further than we have a model of it that consists of the
real numbers as we are thinking about them ("standard model").
We can prove within the theory that the domain (the set of real
numbers) is not countable (Cantor’s theorem).
But by Löwenheim-Skolem, it has a model where the domain is not
countable. (Skolem’s paradox.)
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After the paradoxes
Some consequences
Let us suppose that we have a first-order theory of real numbers
that contains the usual operations and relations for real numbers
and proves at least some simple propositions about them. Let us
suppos e further than we have a model of it that consists of the
real numbers as we are thinking about them ("standard model").
We can prove within the theory that the domain (the set of real
numbers) is not countable (Cantor’s theorem).
But by Löwenheim-Skolem, it has a model where the domain is not
countable. (Skolem’s paradox.)
Not a contradiction; but implies that some important notions (e.g.
countability) are incurably relative, model-dependent.
(Putnam: "Models and reality", 1980)
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After the paradoxes
Consequences continued
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After the paradoxes
Consequences continued
The first-order theory of real numbers has non-standard models,
e.g. models where there are infinitely small positive numbers.
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After the paradoxes
Consequences continued
The first-order theory of real numbers has non-standard models,
e.g. models where there are infinitely small positive numbers.
Let us consider the following set of propositions:
{0 < a < 1, 0 < a < 1/2, . . . , 0 < a < 1/n, . . .}∪{Axioms of the theory}
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After the paradoxes
Consequences continued
The first-order theory of real numbers has non-standard models,
e.g. models where there are infinitely small positive numbers.
Let us consider the following set of propositions:
{0 < a < 1, 0 < a < 1/2, . . . , 0 < a < 1/n, . . .}∪{Axioms of the theory}
Every finite subset of it has a model (namely the standard one
extended by a suitable interpretation of ’a’). Therefore, the whole
set has a model, too, and it is a model of the axioms.
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After the paradoxes
Consequences continued
The first-order theory of real numbers has non-standard models,
e.g. models where there are infinitely small positive numbers.
Let us consider the following set of propositions:
{0 < a < 1, 0 < a < 1/2, . . . , 0 < a < 1/n, . . .}∪{Axioms of the theory}
Every finite subset of it has a model (namely the standard one
extended by a suitable interpretation of ’a’). Therefore, the whole
set has a model, too, and it is a model of the axioms.
Similarly, Peano arithmetics has models with infinitely large
numbers.
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After the paradoxes
Consequences continued
The first-order theory of real numbers has non-standard models,
e.g. models where there are infinitely small positive numbers.
Let us consider the following set of propositions:
{0 < a < 1, 0 < a < 1/2, . . . , 0 < a < 1/n, . . .}∪{Axioms of the theory}
Every finite subset of it has a model (namely the standard one
extended by a suitable interpretation of ’a’). Therefore, the whole
set has a model, too, and it is a model of the axioms.
Similarly, Peano arithmetics has models with infinitely large
numbers.
BTW., nonstandard models of Peano arithmetics can be
characterized by the following 2-order sentence:
∃X(∃xXx∧∀x(Xx → x > 0)∧∀y[∀x(Xx → y < x) → ∀x(Xx → x > y+ )])
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After the paradoxes