Question: What subsets of R x R are definable in (R:<)?

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In summary, the only definable subsets of (R;<) are R and the empty set, since x to x+1 is an automorphism that changes all subsets except for R and the empty set. For the second part of the question, you can define (R,a) and (R,a) for some fixed a, as well as the empty set, since (a,b)<(c,d) is not defined and the order relation is still on the real numbers. The definition of "definable set" is not given in the conversation.
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moo5003
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Homework Statement



What subsets of the real line R are definable in (R;<)? What subsets of the plan R x R are definable in (R:<)?


The Attempt at a Solution



R and the empty set are the only definable subsets of (R;<) since:

x to x+1
Is an automorphism and changes all subsets except for R and the empty set, therefore those subsets are the only possible definable subsets.

R(x) := All x ~(x<x)

ie: All real numbers hold this property

Empty Set (x) := All x (x<x)

ie: Nothing holds this property.

Question: When answering the second part of this question for RxR. I'm not completley sure how you can say (a,b) < (c,d). My answer which I'm a little unsure of right now is that you can define (R,a) and (R,a) for some fixed a. (As well as the empty set). Any help would be appreciated.
 
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  • #2
Perhaps it would help if you defined "definable"! What is the definition of "definable set" you are using?

You can't say (a,b)< (c,d). That's why you problem says "(R: <)". The order relation is still on the real numbers.
 
  • #3
x to x+1 doesn't change the set Z.
 

FAQ: Question: What subsets of R x R are definable in (R:<)?

1. What is the meaning of "Logic + Definability (R,<)"?

"Logic + Definability (R,<)" refers to a specific mathematical theory that combines the concepts of logic and definability in the context of the real numbers (R) and the less-than relation (<). This theory aims to study the logical properties of real numbers and how they can be defined using logical formulas.

2. What is the role of logic in "Logic + Definability (R,<)"?

Logic plays a crucial role in "Logic + Definability (R,<)" as it provides a formal framework for reasoning about the real numbers and their properties. It allows us to make precise statements and proofs about the behavior of real numbers, which can then be used to define and characterize these numbers using logical formulas.

3. How does definability relate to "Logic + Definability (R,<)"?

In "Logic + Definability (R,<)", definability refers to the ability to express a mathematical concept or property using logical formulas. This theory explores the extent to which real numbers can be defined using logical formulas and how these definitions can be used to prove theorems about the real numbers.

4. What are some applications of "Logic + Definability (R,<)"?

"Logic + Definability (R,<)" has a wide range of applications in mathematics, computer science, and other fields. It has been used to study the complexity of algorithms, to prove theorems in number theory and analysis, and to understand the foundations of mathematics. It has also been applied to various problems in logic, including the decidability of theories and the consistency of axiomatic systems.

5. What are some open questions in "Logic + Definability (R,<)"?

Despite significant progress in "Logic + Definability (R,<)", there are still many open questions that researchers are actively investigating. Some of these include the existence of a complete axiomatization for "Logic + Definability (R,<)", the decidability of various extensions of this theory, and the connections between "Logic + Definability (R,<)" and other branches of mathematics such as topology and model theory.

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