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Mathematical Logic Final Solutions

创建日期：2024-07-17 16:58:09

所属分类：数学mathematics

标签： Mathematical

Mathematical Logic Final Solutions

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Mathematical Logic Final Solutions

Use Soundedness Theorem for first-order logic to prove that T1 is independent of the other axioms, that isT2, T3 b T1, T2, T3 b ¬T1

XXX December 14th, 2019

Exercise 3

Use Soundedness Theorem for first-order logic to prove that T1 is independent of the other axioms, that is

T2, T3 b T1, T2, T3 b ¬T1

Proof. We first give an L-structure R such that R € T2, T3 and R B T1. The universe is N w , where w is something other than a natural number. Let

S be the successor function so that S(x) = x + 1, if x N and S(w) = w. Define the relation < by

<= {(a, b)|a, b ∈ R} ∪ {(w, w)}

It is easy to see that R € T2, T3 . Also R B T1 since w < w. Thus by soundedness theorem, we have T2, T3 b T1.

Next we give another L-structure R such that R € T2, T3 and R B T1. Ther universe in this case is simply R, where < is given by the usual inequality and S the successor function. It follows that R € T2, T3 and R B T1 as x < x for all x N. Thus by Soundedness theorem, we have T2, T3 b T1,

which completes the proof.

Exercise

Is T a consistent theory? Justify your answer.

Proof. T is a consistent theory. By completeness theorem it suffices to give a model R for this theory. The universe is the natural numbers N, with < the ususal strict inequality. S is the usual successor function, i.e., S(x) = x + 1 for x N. It is easy to see that all T1, T2, T3 are satisfied. Hence it is a consistent theory.

Exercise 5

Give a T -deduction of x[ Sx < Sx]. Give a full deduction. Name all the logical axioms and inference rules involved in the deduction.

Proof. First note that from T1, we know that x[ x < x]. Since Sx is substitutable for x, we apply Q1 and conclude that Sx < Sx. Now we have shown

∀x[¬x < x] → ¬Sx < Sx

We apply QR and conclude that ∀x[¬Sx < Sx], which completes the proof.

Exercise 6

Sketch a T -deduction of x[ Sx = x]. ame all the logical axioms and infer- ence rules involved in the deduction.

Proof. Note that

¬x < x ∧ (x < Sx) → ¬x = Sx

Here we are using T1 and T3 and the fact that the above statement is a tautology (one can use a Truth table to prove this). Hence we apply PC and QR and conclude that ∀x[¬x = Sx].

Exercise 7

Prove that any model for T is infinite.

Proof. Assume, on the contrary, that there is a finite model. Suppose the universe is some finite set. Let x be an element in this set. Note that by T3, we have x < Sx and Sx < S2x. Since the set is finite, there must exist j, i ∈ N with i =ƒ j such that Six = Sjx. Assume without loss of generality that i < j(here < means the usual order for natural numbers). By T2 andT3, we conclude that Six < Sjx. But since Six = Sjx, we have Six < Six, which contradicts Axiom T1. Therefore any model for T must be infinite.

Exercise 8

Prove that we have

R € φ → N0 € φ

for any purely existential sentence φ.

Proof. Let φ be a purely existential sentence. By the inductive definition, one can prove inductively from N0 axioms that N0 € φ, hence N0 φ by completeness theorem.

Exercise 9

Prove that

N0 b ∀x[x < 6 → (x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 5)]

Proof. We will give another LNT -structure R such that R € N0 but R B

∀x[x < 6 → (x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 5). The

universe is the set of all integers Z with usual addition and standard strict ordering of integers. It is clear that this model satisfy all N0 axioms, namely, R € N0(all purely existential properties that hold in natural numbers also

hold in integers). However, R B ∀x[x < 6 → (x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 5), since x can also be all negativer integers.

Exercise 10

Do we have

N € φ ⇔ N0 € φ

for any purely existential sentence? Justify your answer.

Proof. First suppose N φ for any purely existential sentence φ. By sound- edness theorem, we have R € φ. Then by what we have proved in Problem 8, we have N0 φ.

Next we suppose that N0 φ. By soundedness theorem, we have N0 € φ. Since φ is purely existential, by N0 axioms we conclude that N € φ. Finally, by completeness theorem, we must have N € φ, which completes the proof.

Exercise 11

Do we have

N € φ ⇔ N0 € φ

for any Σ-sentence φ? Justify your answer.

Proof. This is not true. From Problem 9 we have

N0 b ∀x[x < 6 → (x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 5)]

However it is easy to see that N € ∀x[x < 6 → (x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 5)] by a standard N -deduction.

Exercise 12

Does there exist an LNT -structure R such that R € N and R B N0? Justify your answer.

Proof. Yes, there exists one. Let R be one LNT -structure whose universe is N w , where w is an element other than natural numbers. Addition is the usual addition of numbers with a + w = w for all a N w . S is the sucessor function with S(x) = x + 1 if x N and S(w) = w. The relation < is defined by

<= {(a, b)|a, b ∈ N} ∪ {(a, w)|a ∈ N ∪ {w}}

It is easy to see that R € N . However R B N0. For instance, w + 5 = in the usual standard structure, but nevertheless R € w + 5 = w + 3

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