Paradoxes and infinities

[kedaman]

I was thinking about classifying paradoxes, and found that

"This is false"

is a recursive function bool x(){return x()==false} and that it can't be evaluated because it won't terminate the recursion. It reminds me of diverging series but that's another story.

What about "This is true", it's also recursive and it's obviously not considered a paradox, however only as if there is no way to mathematically evaluate you think it's "logical" actually it just "makes sense" to say that it's true because it just seems to work out no matter how many times you recurse. Problem is that infinity is not something mathematical...

So, neither are statements, but paradoxes, what i think

What do you think?

[Guv]

This statement is false.
This statement is true.

The above are self referential statements, which are considered taboo due to their being known to cause logical problems. The first statement is obviously paradoxical. It cannot be classified as either true or false. The second seems ambiguous. Declaring it to be either true or false seems valid, but I like to think of it as true.

Circular definitions are another taboo. The Barber of Seville paradox is due to a circular definition. Modern logic requires some undefined primitive terms to avoid this particular type of paradox.

Hofstadter has published some cute self referential statements which are more amusing than paradoxical. For example.

This English sentence is difficult to translate into French.

For a while I was a Russian sentence, but now I am an English sentence.

The above follow all the rules of English grammar, but seem difficult to grok completely. I do not have his book handy and do not remember any of his others.

I have friends who got into fierce arguments over the first of the above. One insisted on a literal translation (Cette sentence Englais . . .), the other wanted to convey the senses with his translation (Cette sentence Francais . . .).

There are various paradoxes which can be created by improper operations on infinites series. For example the following series converges.

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 . . .

Paradoxical reults can be obtained by expressing it as the difference of two diverging series (n improper operation).

(1 + 1/3 + 1/5 . . .) - (1/2 + 1/4 + 1/6. . .)

[Alphanos]

As people, we might ask whether its true or false, and thus end up in the paradox kedaman is refering to. But at the level computer hardware works at, it isn't really like that, because when there is current (1) is does something, otherwise it doesn't; a lot like VB events really.

[kedaman]

I could well construct a circuit that would try to evaluate one of those, it would go on for ever.

[Alphanos]

I was simply pointing out that at such a low level, its not really necessary to know what something is as long as you know what to do when something is there, if that made sense.

What is 2? Is 2 two? How can you prove it?

[kedaman]

if that made sense

nope, it buggered up resolving the references

What is 2? Is 2 two? How can you prove it?

No, you can't proove anything, only define

[Kzin]

Happy New year Guv and Ked!

. . . and as a New Year's present to Ked I present him with this "On Formally Undecidable Propositions"


I'm still staggering around with a wretched flu' at the moment so won't stay at the keyboard long but I hope it stops the discussion recurring infinitely

[Alphanos]

Public Sub OnlineAtVBForums()
On Error Resume Next
 
Dim i As Long
Dim PostLength As Long
Dim Post As String
Dim PostBytes() As Byte
Randomize Timer
 
Do
     i = 1
     PostLength = CLng(Rnd * 10000)
     ReDim PostBytes(PostLength) As Byte
     Do
         PostBytes(i) = Int(Rnd * 26) + 98
         i = i + 1
     Loop Until i > PostLength
 
     Post = StrConv(PostBytes(), vbUnicode)
     Erase PostBytes()
     PostMessage(Post)
     Post = vbNullString
Loop
End Sub

[kedaman]

Thanks, that was what I was looking for The inperfection of logic

[kedaman]

[Alphanos]

That is a VERY cool link.