15-819/18-879: Logical Analysis of Hybrid Systems
08: First-Order Real Arithmetic
André Platzer
[email protected]
Carnegie Mellon University, Pittsburgh, PA
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André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
1 / 10
Outline
1
First-Order Real Arithmetic
Axioms of Reals
Real-Closed Fields
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
1 / 10
Outline
1
First-Order Real Arithmetic
Axioms of Reals
Real-Closed Fields
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
1 / 10
In hybrid systems, there is significant logical
structure in the properties, the system, the
reasoning, ...
We need to understand the reals first
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
2 / 10
Axioms of Reals
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
11
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
Antisym. ∀x ∀y (x ≥ y ∧ y ≥ x → x = y )
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
11
12
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
Antisym. ∀x ∀y (x ≥ y ∧ y ≥ x → x = y )
Total ∀x ∀y (x ≥ y ∨ y ≥ x)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
11
12
13
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
Antisym. ∀x ∀y (x ≥ y ∧ y ≥ x → x = y )
Total ∀x ∀y (x ≥ y ∨ y ≥ x)
Additive ∀x ∀y ∀z (x ≥ y → x + z ≥ y + z)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
Antisym. ∀x ∀y (x ≥ y ∧ y ≥ x → x = y )
Total ∀x ∀y (x ≥ y ∨ y ≥ x)
Additive ∀x ∀y ∀z (x ≥ y → x + z ≥ y + z)
Positive ∀x ∀y (x ≥ 0 ∧ y ≥ 0 → x · y ≥ 0)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
Axioms of Reals
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
Neutral ∀x x + 0 = x
Inverse ∀x ∃y x + y = 0
Abelian ∀x ∀y (x + y = y + x)
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
Neutral ∀x x · 1 = x
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
Abelian ∀x ∀y (x · y = y · x)
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
Transitive ∀x ∀y ∀z (x ≥ y ∧ y ≥ z → x ≥ z)
Antisym. ∀x ∀y (x ≥ y ∧ y ≥ x → x = y )
Total ∀x ∀y (x ≥ y ∨ y ≥ x)
Additive ∀x ∀y ∀z (x ≥ y → x + z ≥ y + z)
Positive ∀x ∀y (x ≥ 0 ∧ y ≥ 0 → x · y ≥ 0)
Sup “Non-empty subsets with upper bounds have supremum”
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
3 / 10
First-order Axioms of Reals
What is a good set of first-order axioms for the reals R?
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
4 / 10
First-order Axioms of Reals
What is a good set of first-order axioms for the reals R?
Theorem (downward Skolem-Löwenheim’1915-20)
Let Γ be a countable set of first-order formulas.
Γ has a model ⇒ Γ has an infinite countable model
“first-order logic cannot distinguish different infinities”
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
4 / 10
First-order Axioms of Reals
What is a good set of first-order axioms for the reals R?
Theorem (downward Skolem-Löwenheim’1915-20)
Let Γ be a countable set of first-order formulas.
Γ has a model ⇒ Γ has an infinite countable model
“first-order logic cannot distinguish different infinities”
Corollary
The reals cannot be characterized (up to isomorphism) in first-order logic
(nor any other infinite structure really, not even in the generated case)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
4 / 10
First-order Axioms of Reals
What is a good set of first-order axioms for the reals R?
Theorem (downward Skolem-Löwenheim’1915-20)
Let Γ be a countable set of first-order formulas.
Γ has a model ⇒ Γ has an infinite countable model
“first-order logic cannot distinguish different infinities”
Corollary
The reals cannot be characterized (up to isomorphism) in first-order logic
(nor any other infinite structure really, not even in the generated case)
But the first-order “view” of the reals is still fairly amazing
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
4 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
×
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
doesn’t know about π has π
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
×
×
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
doesn’t know about π has π
not closed (lim)
closed
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
×
×
X
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
doesn’t know about π has π
not closed (lim)
closed
dense in R
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
×
×
X
×
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
doesn’t know about π has π
not closed (lim)
closed
dense in R
FOLQ undecidable
FOLR decidable
LAHS/08: First-Order Real Arithmetic
5 / 10
Of Rationals and Reals
Rationals and Reals are . . .
Similar
X
Dissimilar
×
X
×
×
X
×
André Platzer (CMU)
Q
R
similar axioms (ordered field of characteristic 0)
countable
uncountable
algebraic closures indistinguishable algebraically
doesn’t know about π has π
not closed (lim)
closed
dense in R
FOLQ undecidable
FOLR decidable
LAHS/08: First-Order Real Arithmetic
5 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
1
−1 is not a sum of squares in R.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
1
2
−1 is not a sum of squares in R.
P
For every x1 , . . . , xn ∈ R, ni=1 xi2 = 0 implies x1 = · · · = xn = 0.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
1
2
3
−1 is not a sum of squares in R.
P
For every x1 , . . . , xn ∈ R, ni=1 xi2 = 0 implies x1 = · · · = xn = 0.
R admits an ordering that makes R an ordered field (ordering
compatible with +, ·).
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
1
2
3
−1 is not a sum of squares in R.
P
For every x1 , . . . , xn ∈ R, ni=1 xi2 = 0 implies x1 = · · · = xn = 0.
R admits an ordering that makes R an ordered field (ordering
compatible with +, ·).
What’s missing?
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Relaxing Reals to Real Fields
Definition (Formally real field)
Field R is a (formally) real field iff, equivalently:
1
2
3
−1 is not a sum of squares in R.
P
For every x1 , . . . , xn ∈ R, ni=1 xi2 = 0 implies x1 = · · · = xn = 0.
R admits an ordering that makes R an ordered field (ordering
compatible with +, ·).
What’s missing?
That we have enough elements in R.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
6 / 10
Real-Closed Fields
Definition (Real-closed field)
Field R is real-closed field iff, equivalently:
1
R is an ordered field where every positive element is a square and
every univariate polynomial in R[X ] of odd degree has a root in R
(then this order is, in fact, unique).
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
7 / 10
Real-Closed Fields
Definition (Real-closed field)
Field R is real-closed field iff, equivalently:
1
R is an ordered field where every positive element is a square and
every univariate polynomial in R[X ] of odd degree has a root in R
(then this order is, in fact, unique).
2
R is not algebraically
closed but its field
√
extension R[ −1] = R[i]/(i 2 + 1) is algebraically closed.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
7 / 10
Real-Closed Fields
Definition (Real-closed field)
Field R is real-closed field iff, equivalently:
1
R is an ordered field where every positive element is a square and
every univariate polynomial in R[X ] of odd degree has a root in R
(then this order is, in fact, unique).
2
R is not algebraically
closed but its field
√
extension R[ −1] = R[i]/(i 2 + 1) is algebraically closed.
3
R is not algebraically closed but its algebraic closure is a finite
extension, i.e., finitely generated over R.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
7 / 10
Real-Closed Fields
Definition (Real-closed field)
Field R is real-closed field iff, equivalently:
1
R is an ordered field where every positive element is a square and
every univariate polynomial in R[X ] of odd degree has a root in R
(then this order is, in fact, unique).
2
R is not algebraically
closed but its field
√
extension R[ −1] = R[i]/(i 2 + 1) is algebraically closed.
3
R is not algebraically closed but its algebraic closure is a finite
extension, i.e., finitely generated over R.
4
R has the intermediate value property, i.e., R is an ordered field such
that for any polynomial p ∈ R[X ] with a, b ∈ R, a < b and
p(a)p(b) < 0, there is a ζ with a < ζ < b such that p(ζ) = 0.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
7 / 10
Real-Closed Fields
Definition (Real-closed field)
Field R is real-closed field iff, equivalently:
1
R is an ordered field where every positive element is a square and
every univariate polynomial in R[X ] of odd degree has a root in R
(then this order is, in fact, unique).
2
R is not algebraically
closed but its field
√
extension R[ −1] = R[i]/(i 2 + 1) is algebraically closed.
3
R is not algebraically closed but its algebraic closure is a finite
extension, i.e., finitely generated over R.
4
R has the intermediate value property, i.e., R is an ordered field such
that for any polynomial p ∈ R[X ] with a, b ∈ R, a < b and
p(a)p(b) < 0, there is a ζ with a < ζ < b such that p(ζ) = 0.
5
R is a real field such that no proper algebraic extension is a formally
real field.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
7 / 10
Real-Closed Fields beyond the Reals
Example (Real-closed fields)
Real numbers R.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
8 / 10
Real-Closed Fields beyond the Reals
Example (Real-closed fields)
Real numbers R.
Real algebraic numbers Q̄ ∩ R, that is, real numbers in the algebraic
closure of Q, i.e., real numbers that are roots of a non-zero
polynomial with rational or integer coefficients
p(r ) = 0 for some p ∈ Q[X ] \ {0}
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
8 / 10
Real-Closed Fields beyond the Reals
Example (Real-closed fields)
Real numbers R.
Real algebraic numbers Q̄ ∩ R, that is, real numbers in the algebraic
closure of Q, i.e., real numbers that are roots of a non-zero
polynomial with rational or integer coefficients
p(r ) = 0 for some p ∈ Q[X ] \ {0}
Computable numbers, i.e., those that can be approximated by a
computable function up to any desired precision.
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
8 / 10
Real-Closed Fields beyond the Reals
Example (Real-closed fields)
Real numbers R.
Real algebraic numbers Q̄ ∩ R, that is, real numbers in the algebraic
closure of Q, i.e., real numbers that are roots of a non-zero
polynomial with rational or integer coefficients
p(r ) = 0 for some p ∈ Q[X ] \ {0}
Computable numbers, i.e., those that can be approximated by a
computable function up to any desired precision. π lives here!
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
8 / 10
Real-Closed Fields beyond the Reals
Example (Real-closed fields)
Real numbers R.
Real algebraic numbers Q̄ ∩ R, that is, real numbers in the algebraic
closure of Q, i.e., real numbers that are roots of a non-zero
polynomial with rational or integer coefficients
p(r ) = 0 for some p ∈ Q[X ] \ {0}
Computable numbers, i.e., those that can be approximated by a
computable function up to any desired precision. π lives here!
ZFC-Definable numbers, i.e., those real numbers a ∈ R for which
there is a first-order formula ϕ in set theory with one free variable
such that a is the unique real number for which ϕ holds true.
I , β |= ϕ iff I (x) = a
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
8 / 10
The advantages of implicit definition over construction are
roughly those of theft over honest toil [Russell]
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
9 / 10
First-Order Axiom Schemes of Real-Closed Fields
1
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
2
Neutral ∀x x + 0 = x
3
Inverse ∀x ∃y x + y = 0
4
Abelian ∀x ∀y (x + y = y + x)
5
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
6
Neutral ∀x x · 1 = x
7
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
8
Abelian ∀x ∀y (x · y = y · x)
9
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
10
¬∃x1 . . . ∃xn (−1 = x12 + · · · + xn2 ) for any n
11
∀x ∀y (x < y ∧ p(x)p(y ) < 0 → ∃z (x < z < y ∧ p(z) = 0)) for
polynomial p (intermediate value property)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
10 / 10
First-Order Axiom Schemes of Real-Closed Fields
1
Commutative group (R, +): ∀x ∀y ∀z x + (y + z) = (x + y ) + z
2
Neutral ∀x x + 0 = x
3
Inverse ∀x ∃y x + y = 0
4
Abelian ∀x ∀y (x + y = y + x)
5
Commutative group (R \ {0}, ·): ∀x ∀y ∀z x · (y · z) = (x · y ) · z
6
Neutral ∀x x · 1 = x
7
Inverse ∀x (x 6= 0 → ∃y x · y = 1)
8
Abelian ∀x ∀y (x · y = y · x)
9
Distributive ∀x ∀y ∀z (x · (y + z) = (x · y ) + (x · z))
10
¬∃x1 . . . ∃xn (−1 = x12 + · · · + xn2 ) for any n
11
∀x ∀y (x < y ∧ p(x)p(y ) < 0 → ∃z (x < z < y ∧ p(z) = 0)) for
polynomial p (intermediate value property)
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
10 / 10
André Platzer (CMU)
LAHS/08: First-Order Real Arithmetic
10 / 10