Prove the following theorems in 𝒩 : . Prove that the system 𝒩 is complete in the sense of 2509.

Name: Problem 5, Chapter 3: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)
Uploaded: 2019-08-08T01:48:02-08:00
Duration: 6 min 48 s
Channel: Brian Ketelobeter
Description: Prove the following theorems in 𝒩 : . Prove that the system 𝒩 is complete in the sense of 2509.

step 1 Good day. Today we're going to be looking at problem number 36. And again, I didn't really bothe

Prove the following theorems in 𝒩 : . Prove that the system...

Key Concepts

Formal Systems

A formal system is a structured framework composed of a formal language, axioms, and inference rules that allow for the derivation of theorems. It provides a rigorous basis for reasoning and proving statements by ensuring that every step in a proof adheres to clearly defined syntactical rules. In the context of arithmetic, the system ? is such a framework aimed at capturing the properties and operations of natural numbers.

Axiomatization of Arithmetic

Axiomatization involves specifying a minimal set of fundamental assumptions or axioms that define the arithmetic of natural numbers. Systems like ? are built on axioms that describe properties such as zero, the successor function, and the principle of induction, thereby giving a complete abstract characterization of natural number arithmetic.

Induction Principle

The induction principle is a critical axiom in the axiomatization of arithmetic. It states that if a property holds for zero and holds for the successor of any number for which it holds, then the property holds for all natural numbers. This principle is essential not only for proving properties about numbers but also plays a central role in establishing the completeness of an arithmetic system.

Completeness Theorem

The completeness theorem, in the context of formal systems, asserts that every semantically valid statement (one that is true in every model of the system) is syntactically provable within the system. Proving completeness for a system like ? means demonstrating that if a formula is true under all interpretations given by the axioms of arithmetic, then there exists a derivation within ? that establishes the formula as a theorem. This concept bridges the gap between semantic truth and syntactic provability.

Transcript

00:01 Good day.

00:03 Today we're going to be looking at problem number 36.

00:07 And again, i didn't really bother writing out the whole problem, but basically the summary is, you know, we have a, we have a non -homogeneous system here.

00:25 We have a non -homogeneous system, and we have a homogeneous system.

00:30 And we're assuming we have a, we have a, we have a, we have a, we have a, fundamental solution set here and that xp is a particular solution and what we want to do is show any solution can be written any solution can be written of this form okay so um okay and now the trick here okay so this is what this is our goal now what do we want to do okay so so basically what we do is we assume we have any two solutions to 36.

01:08 Okay.

01:09 So we have any two solutions to the non -homogeneous ode.

01:18 And this implies, of course, i'm just rewriting both of this.

01:24 That z1 of t and z2, well, the derivatives are equal to, of course, here.

01:34 And now this implies what? well, what implies is if i subtract the two of these expressions, so if i take the difference of these two expressions, what do i get? well, when i take the difference, the f of t is going to cancel.

01:57 So i'm going to get this.

02:00 I'm going to get, well, yeah, i'm going to get this here.

02:04 And that leads me to this expression here.

02:09 Now, a is a matrix, and z2 of t and z1 of t are vectors.

02:19 So because of this, i can get this expression here.

02:24 This is just, again, in case you guys have forgotten.

02:31 I hope you haven't.

02:32 But what this is just a x plus a y is just a x plus y is just a x plus y okay so that's all i'm doing it just looks kind of vamp up because it's got the twos and the t's and everything in here that's really all i'm doing um okay so now of course z2 of t minus z one of t is a vector and so what does that mean? well, of course, if we look at the difference here, i just define zt to be the difference...

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