Decidability of -1<1/0<1 using the ordered field axioms and first order logic

In summary, the conversation discusses the decidable nature of the statement -1<1/0<1 using ordered field/real number axioms and first-order logic. The individual has attempted to prove the statement to be either true or false, but has had no success due to the axioms and theorems only applying to existing objects. The question is whether it is possible to prove the statement or if it is fundamentally undecidable in the given system. Another possibility is to derive a contradiction regardless of the truth value of the statement. The individual also brings up the concept of nonexistent objects and whether statements about them are inherently false. They ask for confirmation on a specific formula and whether it sheds light on the issue. The conversation also touches
  • #1
ad infinitum
17
1
Is the statement -1<1/0<1 decidable using the ordered field/real number axioms and first order logic? I have tried to prove that the statement is either true or false but have had no success since the axioms and theorems only make statements about objects that exist and do not give any clear way to treat those that do not. So I would like to know if it is even possible to prove that the statement is true or false or if it is fundamentally undecidable in the given system. Or alternatively is it possible to derive a contradiction regardless of whether the statement is true or false a la the liar paradox.
 
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  • #2
If you assume that 1/0 acts just like the inverse of any other number, then you can show that it cannot exist in any field. (Note that then -1 < 1/0 < 1 is false because the number you called 1/0 doesn't exist)

Assume that 0 * x = 1 (i.e., that x = 1/0).

Now we also have that for any number in our field r: r * 0 * x = 0 * x = 1. Also, for all r (even when r is 0), we have (r / r) * 0 * x = 0 * x = 1/r. Which gives us that 1 = 1/r for all r. Since every number has an inverse, including 1/0, we have 1 = 1/(1/0) = 0. Therefore, we have only 1 number in our algebra. Other than not being a field, then is fairly uninteresting. Also, the statement x < y is always false since for any x and y in our algebra, x = y.
 
  • #3
Looking back on my first post I now see that I did not properly articulate what it was that I was trying to inquire about. I will be more carful next time.

LukeD said:
If you assume that 1/0 acts just like the inverse of any other number, then you can show that it cannot exist in any field. (Note that then -1 < 1/0 < 1 is false because the number you called 1/0 doesn't exist)

I already understand why 1/0 cannot exist in a field, I simply chose it as an example of a nonexistent object when what I am really interested in is the more general case of whether all statements about nonexistent objects are inherently false. While this seems obvious intuitively, I am not sure how derive it rigorously from the given axioms. I am wondering if there is such an axiom in first order logic that simply asserts this, but I am having trouble understanding the Wikipedia article on the topic. I believe that the following formula might shed some light on this issue and am asking for confirmation.

[tex]Z(t)\rightarrow(\exists xZ(x))[/tex]
where the expression Z(x) stands for any well-formed forumula with the additional convention that Z(t) stands for the result of substitution of the term t for x in Z(x).
 
  • #4
ad infinitum said:
I believe that the following formula might shed some light on this issue and am asking for confirmation.

[tex]Z(t)\rightarrow(\exists xZ(x))[/tex]
where the expression Z(x) stands for any well-formed forumula with the additional convention that Z(t) stands for the result of substitution of the term t for x in Z(x).

If t does not exists, then depending on your interpretation of Z(t), it is either not considered a statement, or it is interpreted as [tex]\exists t Z(t)[/tex], which is false.

Try reading this Wikipedia article: http://en.wikipedia.org/wiki/Definite_description
 
  • #5
If it helps, the following statement is provable from the ordered ring axioms:
For all x: 0x = 1 implies -1 < x < 1.​
 
  • #6
LukeD said:
If t does not exists, then depending on your interpretation of Z(t), it is either not considered a statement, or it is interpreted as [tex]\exists t Z(t)[/tex], which is false.
What I am interested in knowing is which of these interpretations is part of first-order logic. Or are you saying that first-order logic does not itself specify which of these two interpretations to use and that that decision is left up to a particular model.

What about the particular case of mathematical analysis which I believe is based on second-order logic? In order for the definition of a limit to mean what we want it to mean, statements about nonexistent objects must be false. (I think) So where does this rule derive from? Does second-order logic make a judgment on the matter of which interpretation to use or does it come from some other foundation of mathematical analysis?
 

1. What is the significance of the -1<1/0<1 statement in the context of ordered field axioms and first order logic?

The statement -1<1/0<1 refers to the existence of a number between -1 and 1 that is greater than 0. This statement is significant in the context of ordered field axioms and first order logic because it is used to prove the decidability of statements in these mathematical systems.

2. How does the concept of decidability relate to the statement -1<1/0<1?

The concept of decidability refers to the ability to determine the truth or falsehood of a statement in a mathematical system. In this case, the statement -1<1/0<1 is decidable using the ordered field axioms and first order logic, meaning it can be proven to be either true or false within these systems.

3. Can the statement -1<1/0<1 be proven using other mathematical systems?

Yes, the statement -1<1/0<1 can also be proven using other mathematical systems, such as real analysis or abstract algebra. However, the specific methods and axioms used may differ from those used in ordered field axioms and first order logic.

4. What are the implications of the statement -1<1/0<1 being decidable?

The decidability of the statement -1<1/0<1 has implications for the consistency and completeness of the mathematical systems in which it is proven. It also allows for further exploration and development of these systems, as well as applications in various fields, such as computer science and physics.

5. Are there any limitations to using the ordered field axioms and first order logic to prove the statement -1<1/0<1?

While the ordered field axioms and first order logic are powerful tools in proving the decidability of statements, they may not be applicable to all mathematical systems or statements. Additionally, there may be other methods or axioms that can also be used to prove the statement -1<1/0<1, but have not been discovered yet.

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