Comment by cmonnow | Hacker News Reader

Comment by cmonnow | Hacker News Reader

I never understood the fascination people have with self-referential 'paradoxes' like : "This statement is False".

Pronouns do not have an independent existence. Until the pronoun 'This' resolves to an actual statement which has a valid true/false property, it is a recurrence without termination i.e. a infinite loop, that has no meaning.

Suppose I say: 1. Sky is blue. 2. Previous statement is True. 3. Previous statement is True.

And ask you 'Is Statement 3 is True or False?'. You would say 'First I need to know what 'previous' refers to', so you would do pronoun substitution for the word 'Previous' in Statement 3:

3. 'Previous statement is True' is True.

Since this statement has an unresolved pronoun, you would do pronoun substitution for the word 'Previous' in Statement 2:

3. ''Sky is blue' statement is True' is True.

Since there are no more unresolved pronouns, you would evaluate the statement 'Sky is blue', and then the next Statement 2, and finally Statement 3, and say 'Yes, Statement 3 is True'. So far, so good ?

Now, suppose I say:

1. This statement is False.

And ask you 'Is Statement 1 True or False?'. You would say 'First I need to know what 'This' refers to', so you would do pronoun substitution for the word 'This' in Statement 1:

1. 'This statement is False' statement is False.

Since this statement has an unresolved pronoun ('This'), you would do pronoun substitution again.

- ''This statement is False' statement is False' statement is False.

Since this statement has an unresolved pronoun ('This'), you would do pronoun substitution once more.

- '''This statement is False' statement is False' statement is False' statement is False'

And you will keep doing pronoun substitutions forever since statements with unresolved pronouns cannot have a True/False property.

I find this teenage-girl-at-a-rock-concert-feigned-fascination with self-referential 'paradoxes' silly.

I think you're too quick to dismiss what is widely considered to be one of the most important and profound results in logic and computability without taking the time to fully appreciate it.

Your procedure of repeated substitution in order to resolve a paradox does little to shed light on the situation since at the heart of the halting problem is that there's no formal system that can determine whether this repeated substitution will ever come to an end.

Sure in your trivial example we could prove that it never comes to an end and determine that the statement is a paradox and label it as such, but there are some statements where it's not clear whether there's a paradox in the first place and there are some statements that are a paradox in one interpretation but in another interpretation are perfectly sensible.

Rest assured, no mathematician is scratching their head wondering whether "This statement is false." is some kind of mysterious statement whose undecidability has profound consequences. The issue is with statements like whether there exist 3 integers, x, y, z such that:

x^3 + y^3 + z^3 = 114

That statement in and of itself may very well be a "paradox" similar to a statement such as "This statement has no proof in theory T." even though on the surface it looks like a perfectly reasonable equation that should either have a solution or not have a solution. I mean either three such numbers exist or they don't exist, right?

And yet... it's possible that for the equation I gave there are solutions only under some interpretations of what we normally call natural numbers and in other interpretations of what we call natural numbers there isn't a solution, and there's no formal system that can filter out one interpretation over another so that there is one and only one unambiguous interpretation of natural numbers that we can always rely upon as the "real" interpretation.

There are perfectly normal looking sentences and problems that on the surface don't appear at all to be paradoxes or self referential, but then become so when you try to pin the question more precisely.

That's where the fascination comes from, from looking at a seemingly normal looking equation or statement about numbers that should just be true or false and realizing that whether it's true or false depends on some very deep and as-of-yet unknown properties of what we even mean when we talk about natural numbers.

PP seems to be asking for an example that is not obviously self-contradictory yet demonstrates a serious, unsurmountable problem, thereby illustrating the incompleteness theorem.

Is the theorem you describe such an example? If so, you have refuted PP with a non-handwaving result.

If not, then you've provided more evidence supporting PP's complaint.

People have a fascination with "This sentence is false" because it highlights the need to be precise distinguishing between the language of a logical system and a metalanguage to talk about the language. That discovery or 'invention' if you will is something that was not put into rigorous foundation until the 20th century.

Foundational theories of Computer Science would be rendered

inconsistent if they allowed the construction of the

proposition I'mFalse or the [Gödel 1931] proposition

I'mUnprovable using fixed points.

    Fortunately, orders on propositions prohibit the 
    construction of the proposition *I'mFalse* and also 
    prohibit the construction of the [Gödel 1931] 
    proposition *I'mUnprovable*.

Well, the liar paradox dates to antiquity, the halting problem has been well known for 70+ years, etc., and yet.... it turns out we can still use logic and software to solve lots of practical problems anyway, simply by avoiding self-referential contradictions and infinite loops.

Arithmetic seems to have a number (so to speak) of practical applications as well.

Perhaps it's a bit like theory vs. practice, or maybe mathematics vs. engineering.

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