Set of representable real numbers

• Office_Shredder
In summary, the conversation discusses the countability of the class of all real numbers that can be represented as a sentence in logic and whether it is a set under the standard ZF axioms. There is also a discussion about the possibility of defining this class using first-order logic statements and attempts at proving that the definable real numbers do form a set.
Office_Shredder
Staff Emeritus
Gold Member
From what I understand, the class of all real numbers that we can represent as a sentence in logic is countable. But I'm not sure if it's a set under the standard ZF axioms... it seems intuitive that it should be, since the axioms are really designed to prevent problems involving sets that are too large, and countable sets are fairly innocent, but was wondering if anyone knew one way or the other.

Well, every subclass of a set is a set, according to the axiom of separation, but the question is whether the set of definable/representable real numbers is a class, i.e. can be defined by a formula of set theory.

Here's my attempt at a proof: the class of all formulas of set theory is clearly a set (it can easily be defined inductively). Now if there is a class function f: formula phi -> the subset of R defined by phi, then (because the image of a set is a set) the class of all definable subsets of R is a set. Being a singleton set is a definable condition, so we take the intersection of this class and the class of singletons. We take the union of this set to get the set of definable real numbers. So we need to construct the function f, but this can be done inductively (unless I'm mistaken). Therefore, the definable reals form a set.

Well, every subclass of a set is a set, according to the axiom of separation

Only if they can be described as satisfying a first order logic statement right? According to wikipedia, no such statement exists for the definable reals.

Yes, which is what I said and then went on to attempt a proof without using the presupposition that the definable numbers form a class.

ETA: every class is definable by a first-order formula of set theory, so your "only if they can be described as satisfying a first order logic statement" is merely a restatement of the assumption that it's a class.

ETA 2: to try to make the construction of f explicit: $f(x \in a) = a, f(\neg\varphi)=\neg f(\varphi),f(\varphi \wedge \psi) = f(\varphi) \cap f(\psi), f(\exists y \varphi(x,y))=\cup_{y \in V} f(\varphi(x,y))$

The quantifier step doesn't work as it is, I can't figure out how to fix it right now, but intuitively speaking it should be possible, as first-order logic can be immersed into set theory.

Last edited:
I think the problem is we're defining class differently. I think I see what you're saying there

You're looking at:

{x in R : there exists phi s.t. "phi is a formula" and "there exists unique y in R s.t. V models phi[y]" and "V models phi[x]"}

The condition for x to be in this set doesn't look like a formula of ZFC, but you can probably arithmetize the language such that it becomes equivalent to a formula of ZFC, e.g. (there exists phi s.t. "phi is a formula") is equivalent to (there exists n s.t. n is in the set of Godel numbers of ZFC formulas) where the set of Godel numbers of ZFC formulas is somehow definable.

1. What is a set of representable real numbers?

A set of representable real numbers is a collection of all the numbers that can be expressed as a decimal number, including both rational and irrational numbers. It is a fundamental concept in mathematics and is essential in understanding the real number system.

2. How is a set of representable real numbers different from the set of all real numbers?

The set of all real numbers includes both representable and non-representable numbers, such as imaginary numbers. A set of representable real numbers only includes numbers that can be expressed as decimal numbers, making it a subset of the set of all real numbers.

3. Can infinite numbers be included in a set of representable real numbers?

Yes, a set of representable real numbers can include infinite numbers, such as repeating decimals like 0.3333... or non-repeating decimals like pi (3.141592...). These numbers can be expressed as decimal numbers and therefore, are considered representable.

4. How is a set of representable real numbers useful in scientific research?

A set of representable real numbers is useful in scientific research as it provides a framework for understanding and analyzing numerical data. It also allows scientists to make accurate calculations and measurements, which are essential in various fields of science, including physics, chemistry, and engineering.

5. Are all real numbers included in a set of representable real numbers?

Yes, all real numbers are included in a set of representable real numbers. This is because any number that can be expressed as a decimal number, no matter how large or small, is considered a real number and can be included in the set of representable real numbers.

• Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
• Topology and Analysis
Replies
2
Views
653
• Set Theory, Logic, Probability, Statistics
Replies
8
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
11
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
4
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
• Calculus
Replies
24
Views
3K
• Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K