Thread for Non-CA Academic Questions

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Re: Thread for Non-CA Academic Questions

Post by fluffykitty » June 18th, 2019, 8:56 pm

Moosey wrote:Came across this by defining a different Ord-type function:
if I define Ξ (Greek letter Xi) as the smallest (?) ordinal such that n= w^((w^n)2)
w^(w^n*2)=(w^w^n)^w^n. Assuming n=e_x, (w^w^e_x)^w^e_x=e_x^e_x>e_x. If n≠e_x for any x, then n<w^n<w^n*2<w^(w^n*2) so Ξ can be neither epsilon nor nonepsilon, and therefore is illdefined.
Moosey wrote:Can someone explain inaccessible cardinals? Apparently they are κs such that κ= an uncountable κ such that it is the ב_κ th regular cardinal—
Just "uncountable κ that is the κth regular cardinal" works.
Moosey wrote:What’s a regular cardinal?
I know that ב_n+1 = 2^(ב_n) so I sorta understand that.
A cardinal such that any sequence of lower cardinals that exceeds any cardinal less than κ at some point must have length at least κ.
Moosey wrote:The other thing I understand is that κ is 1-inaccessible if it is the κth inaccessible cardinal, and I guess it’s probably n+1 inaccessible if it’s the κth n-inaccessible cardinal.
The other other thing I get is that κ is hyper-inaccessible if it’s κ-inaccessible.
The other other other thing I think I get is that κ is 1-hyper-inaccessible if it’s the κth hyper-inaccessible number, and
it’s n+1-hyper-inaccessible if it’s the κth n-hyper-inaccessible number.
I love the jargon at the end of this page:
http://cantorsattic.info/Inaccessible#D ... essibility
cantor’s attic wrote:κ is hyper^a-inaccessible if it is hyper-inaccessible and for every b<a it is κ-hyper^b-inaccessible where κ is a-hyper^b-inaccessible if it is hyper^b-inaccessible...
I suppose the logical continuation is that κ is actually-hyper-hyper-inaccessible if it’s hyper^κ inaccessible, whether that’s even possible or not.
You can't prove it won't happen (unless you have an axiom stating that inaccessibles do not exist or something like that).
Moosey wrote:EDITLATERTHANTHERESTOFPOST:
Oh, I get it— inaccessible numbers are to aleph_n what epsilon numbers are to w^n:
All inaccessible numbers κ = aleph_κ.
That works too, but it's much larger than just nesting aleph_κ w times on 0.
Moosey wrote: </late edit>
Also, can somebody describe (cue laughtrack) indescribable cardinals? (I don’t expect you to speak of the ineffables, of course.)
Cardinals such that you can't describe them without also describing a bunch of lower cardinals (for a certain definition of "describe")
Moosey wrote: Also reflection and regular cardinals and pretty much everything else building up to indescribables, in a comprehensible way?
https://googology.wikia.org/wiki/User_b ... ion_to_OCFs
Moosey wrote: Speaking of comprehensibility, someone should define incomprehensible cardinals. I’ll take a crack at it maybe...

Attempt I:
An incomprehensible cardinal is a cardinal κ such that κ is the κth compressed-κ-inaccessible cardinal
A compressed-n-inaccessible cardinal is defined as follows:
A compressed-0-inaccessible cardinal is a hyper^κ-inaccessible cardinal.
A compressed-n+1-inaccessible cardinal κ is the κth compressed-n-inaccessible cardinal

Inc_0 (the 1st incomprehensible cardinal) is the Inc_0th compressed-Inc_0-inaccessible cardinal.
Weaker than Mahlos. It's not very easy to create an interesting and original large cardinal property (otherwise the Cantor's Attic list would be overflowing with large cardinal axioms)

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Re: Thread for Non-CA Academic Questions

Post by Moosey » June 19th, 2019, 1:14 pm

fluffykitty wrote:
Moosey wrote:Came across this by defining a different Ord-type function:
if I define Ξ (Greek letter Xi) as the smallest (?) ordinal such that n= w^((w^n)2)
w^(w^n*2)=(w^w^n)^w^n. Assuming n=e_x, (w^w^e_x)^w^e_x=e_x^e_x>e_x. If n≠e_x for any x, then n<w^n<w^n*2<w^(w^n*2) so Ξ can be neither epsilon nor nonepsilon, and therefore is illdefined.
What about, say ρ_0,2?
Is that illdefined?

ρ_a,b= the ath ordinal n such that n=(w^n)b


Also, is n=w_n comparable to the 1st inaccessible cardinal?

Is it true that all cardinals are also ordinals?
In that case, is it possible to take, say ψ(I) where ψ is the function used to make the Bachmann–Howard ordinal?
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Re: Thread for Non-CA Academic Questions

Post by fluffykitty » June 20th, 2019, 10:49 am

See next, yes, no, yes, yes but it's not any bigger (you need a more complex OCF for that)

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Re: Thread for Non-CA Academic Questions

Post by Moosey » June 20th, 2019, 11:37 am

fluffykitty wrote: yes but it's not any bigger (you need a more complex OCF for that)
I should hope ψ(I) is not bigger than I, since that’s not the purpose of ψ(n).
It it still uncountable though?
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Re: Thread for Non-CA Academic Questions

Post by fluffykitty » June 20th, 2019, 12:31 pm

*any bigger than the BHO. Also, psi(a)=either psi_0(a) or psi_W(a) depending on your OCF is always countable.

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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 8th, 2019, 10:12 am

How would I make an array notation?
Say, a ggσM array:

{x} = ggσM(x)
{x,y} = ggσM({x,y-1})
{#,1} = {#}
But what happens past that?
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Re: Thread for Non-CA Academic Questions

Post by testitemqlstudop » July 8th, 2019, 11:32 pm

something like

{x, y, z} = {ggom({x,y, z-1}), ggom({x, y})}
{x, y, 1} = ggom({x, y})

{a, b, ... z} = {ggom({a, b, ... z-1}), ggom({a, b, ..., y-1, z}), ... , ggom({a, b, c-1, ..., y, z}), ggom({a, b, ...y})}
{a, b, ... 1} = ggom({a, b, ... y})

can suit you

notice that in the {a ... z} expansion there are only 25 terms, but in each ggom call there are 26 terms, except for the last one which has 25

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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 9th, 2019, 7:51 am

testitemqlstudop wrote:something like

{x, y, z} = {ggom({x,y, z-1}), ggom({x, y})}
{x, y, 1} = ggom({x, y})

{a, b, ... z} = {ggom({a, b, ... z-1}), ggom({a, b, ..., y-1, z}), ... , ggom({a, b, c-1, ..., y, z}), ggom({a, b, ...y})}
{a, b, ... 1} = ggom({a, b, ... y})

can suit you

notice that in the {a ... z} expansion there are only 25 terms, but in each ggom call there are 26 terms, except for the last one which has 25
It would be better to say {x,y,z}=ggσM({ggσM({x,y,z-1}),y,z-1},ggσM({x,y,z-1},y,z-1} for a tad more power
Or, in general, for a 1d array, the entry that is n entries from the last one becomes ggσM({*,y,z-1},y,z-1} stacked on the star n times, where the nth * is x. Repeat for all entries.
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Re: Thread for Non-CA Academic Questions

Post by toroidalet » July 11th, 2019, 1:34 pm

Is it decidable whether 2 context-free grammars generate 2 strings with the same length? (w1∈L1,w2∈L2;|w1|=|w2|)

EDIT: It seems like no, but I don't know how to formalise using the possible increases of length from each nonterminal into an algorithm.
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Re: Thread for Non-CA Academic Questions

Post by testitemqlstudop » July 14th, 2019, 5:56 am

How do I efficiently enumerate all n-bit polyplets?

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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 14th, 2019, 5:53 pm

Moosey wrote:How would I make an array notation?
Say, a ggσM array:

{x} = ggσM(x)
{x,y} = ggσM({x,y-1})
{#,1} = ggσM({#})
{ent#n, ent#n-1, ent#n-2... ent#2, ent#1} = ggσM({...{ent#n,ent#n-1...,(ent#1)-1},{ent#n,ent#n-1...,(ent#1)-1}),{ent#n,ent#n-1...,(ent#1)-1}...(ent#1)-1},{ent#n, ent#n-1, ent#n-2...,(ent#1)-1}..., (ent#1)-1}
I.e. {...,{{{the array with the last element -1}}},{{the array with the last element -1}, {the array with the last element -1},{the array with the last element -1}...the last element-1},{the array with the last element -1}, the last element -1} where no array is larger than the initial one in length
But what happens past that? How do I expand to 2D?
Just to help you undertand what I mean there:

Code: Select all

{4,4,4,4} = ggσM{{{{4,4,4,3},{4,4,4,3},{4,4,4,3},3},{{4,4,4,3},{4,4,4,3},{4,4,4,3},3},{{4,4,4,3},{4,4,4,3},{4,4,4,3},3},3},{{4,4,4,3},{4,4,4,3},{4,4,4,3},3},{4,4,4,3},3}
Since every array ends in an n smaller than that in the original value, it is finite.
It reduces to ggσM^a value requiring ggσM stacked some incredible amount of times to express(the first value in the array) with

Obviously 2D arrays look like this: {a,b,c...[1]d,e,f}
I need to make arrays with arbitrarily large dimensions, like this, for instance.
{a[1]b[2]c[1]d[3]...}

But what I’m wondering is how to define the array for multiple dimensions.
@fluffykitty or some other googologist!
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Re: Thread for Non-CA Academic Questions

Post by toroidalet » July 14th, 2019, 6:56 pm

The main problem I can see is that there is no way to decrease the last number in the array (every number is replaced by one larger than it, unless ggσM(x) is defined suitably) and therefore no way to decrease the number of elements. This can be fixed by making {a1,a2,a3,a4,...an} evaluate to {{{a1,a2,a3,a4,...an} n-1 times,an-1} n-1 times,an-1}. (You could increase it some more by replacing the function of an with some other halting sequence like an encoding of a Goodstein sequence or TREE(n) or something else, as well as nesting arrays inside the nested arrays).

For example, with ggσM(x)=x+1, {4,4} evaluates to

Code: Select all

{4,4}->
{{4,3},3}+1->
{{{{4,2},2}+1,3}+1-> ({x,2}->x+4)
{{{8,2},2}+1,3},3}+1->
{{{12,2}+1,3},3}+1->
{{17,3},3}+1->
{{{17,2},2}+1,3}+1->
{{21,2}+1,3}+1->
{26,3}+1->
{{26,2},2}+2->
{30,2}+2->
36->

For 2d arrays one could interpret them as arrays of arrays in the obvious way or do an array function of the array functions of all the subsets of the square array (in every order) in some order (for n-d arrays, it would be an (n-1)-d array of all possible (n-1)-d hypercubes, and so forth) but there's probably some other way.
Last edited by toroidalet on July 14th, 2019, 7:13 pm, edited 1 time in total.
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 14th, 2019, 7:13 pm

toroidalet wrote:The main problem I can see is that there is no way to decrease the last number in the array (every number is replaced by one larger than it, unless ggσM(x) is defined suitably) and therefore no way to decrease the number of elements. This can be fixed by making {a1,a2,a3,a4,...an} evaluate to {{{a1,a2,a3,a4,...an} n-1 times,an-1} n-1 times,an-1}. (You could increase it some more by replacing the function of an with some other halting sequence like an encoding of a Goodstein sequence or TREE(n) or something else, as well as nesting arrays inside the nested arrays).

For example, with ggσM(x)=x+1, {4,4} evaluates to

Code: Select all

{4,4}->
{{4,3},3}+1->
{{{{4,2},2}+1,3}+1-> ({x,2}->x+4)
{{{8,2},2}+1,3},3}+1->
{{{12,2}+1,3},3}+1->
{{17,3},3}+1->
{{{17,2},2}+1,3}+1->
{{21,2}+1,3}+1->
{26,3}+1->
{{26,2},2}+2->
{30,2}+2->
36->

For 2d arrays one could interpret them as arrays of arrays in the obvious way, but there's probably some other way.
Pardon me, but the way I defined it (with the generous assumption that I typed everything right), the last number in {4,4,4,4} becomes 3, and all terminal 1s are removed. In all instances of the array function being called inside, the arrays are at most as long as the initial array and, if they are as large as it, end in a value less than that of the initial array.

Admittedly I may have typed something wrong when I wrote {4,4,4,4}, but the last value in an array is always decremented.
{4,4,4} ->
ggσM({{{4,4,4,3},{4,4,4,3},{4,4,4,3},3}, {4,4,4,3}, 3}), if {4,4,4} is any simpler.
There are 3 distinct parts of the whole array: first, {{4,4,4,3},{4,4,4,3},{4,4,4,3},3}, then, {4,4,4,3}, then 3.

I defined ggσM at the bottom of this post. It is ~ w^3 + w + 1 in the FGH.
Last edited by Moosey on July 14th, 2019, 7:16 pm, edited 1 time in total.
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Re: Thread for Non-CA Academic Questions

Post by toroidalet » July 14th, 2019, 7:16 pm

I must have misunderstood your definition, in particular thinking that it was {1,2,3,4}->{{{1,2,3,3},{1,2,3,3},{1,2,3,3},{1,2,3,3}},{{1,2,3,3},{1,2,3,3},{1,2,3,3},{1,2,3,3}},{{1,2,3,3},{1,2,3,3},{1,2,3,3},{1,2,3,3}}-1}.
Last edited by toroidalet on July 14th, 2019, 7:18 pm, edited 1 time in total.
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 14th, 2019, 7:17 pm

toroidalet wrote:I think I misunderstood your description; notice that my "correction" is exactly the same as your definition.
Yeah, my definition was really confusing-I could have worded it better.
Also, I edited my post to show what ggoM (for short) is.
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Re: Thread for Non-CA Academic Questions

Post by testitemqlstudop » July 14th, 2019, 9:18 pm

testitemqlstudop wrote:How do I efficiently enumerate all n-bit polyplets?

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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 15th, 2019, 7:11 am

toroidalet wrote: For 2d arrays one could interpret them as arrays of arrays in the obvious way or do an array function of the array functions of all the subsets of the square array (in every order) in some order (for n-d arrays, it would be an (n-1)-d array of all possible (n-1)-d hypercubes, and so forth) but there's probably some other way.
What strategy yields fastest possible growth?
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 16th, 2019, 10:48 am

Moosey wrote:Can anyone generalize tilde-arrow notation’s definition to an arbitrary sequence of tildes and arrows?
Tilde-arrow notation can be found in this post under my Naive attempt to extend Knuth Up Arrow notation:
viewtopic.php?f=12&t=4004&p=79634#p79634

I mean, tilde-arrow notation (TAN? I should make a bunch of functions/notations whose acronyms happen to be colors!) is already a good bit stronger than Knuth Uparrow notation; it can easily express values larger than the smaller googologisms like g_64 (3^~65> G_64), though really Uparrow notation is only a recursive step away from a reasonable expression of g_64 (3^sup3^sup3... is stretching it a bit)
But fortunately TAN goes a good bit beyond that.

Anyways, what I want is a simple recursive definition for what happens if you add a tilde or arrow in front or on the back or whatever. (I kinda implied a definition but I have no idea how to express it.)


Also, what is the limit of TAN?
w^n for a finite n?
w^w?
Possibly w^^n for small, finite n?
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 16th, 2019, 11:10 am

testitemqlstudop wrote:How do I efficiently enumerate all n-bit polyplets?
The best I can think of is to use 0-7 to represent a direction in the Moore neighborhood:

Code: Select all

0 1 2
3 x 4
5 6 7
And then count in base 8.
You have n digits and you throw out the polyplets that are smaller than n cells.

So, 000
Corresponds to:

Code: Select all

x . .
. x .
. . x
And 111:

Code: Select all

x
x
x
As for removing duplicates, you can simply avoid generating the polyplets by checking whether your octal number is "reducible" to a lower number.
However that still wouldn’t throw out all the duplicates.
Instead, you could take all of the output, and, starting from one end, check every polyplet after polyplet A to see whether any match A under reflection and rotation. If polyplet B matches polyplet A, B is thrown out.
Once you get through the line, you go back to just past the first polyplet matching A (which will always be A) and start with the polyplet after A.
However I believe there are some polyplets that wouldn’t be enumerated, such as this one:

Code: Select all

There is no path that a chess king can take in five moves so that he lands on these five cells in any order.
x . x
. x .
x . x
Or, for that matter:

Code: Select all

x . x
. x .
x . .
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 19th, 2019, 1:24 pm

Bump
Moosey wrote:
Moosey wrote:Can anyone generalize tilde-arrow notation’s definition to an arbitrary sequence of tildes and arrows?
Tilde-arrow notation can be found in this post under my Naive attempt to extend Knuth Up Arrow notation:
viewtopic.php?f=12&t=4004&p=79634#p79634

I mean, tilde-arrow notation (TAN? I should make a bunch of functions/notations whose acronyms happen to be colors!) is already a good bit stronger than Knuth Uparrow notation; it can easily express values larger than the smaller googologisms like g_64 (3^~65> G_64), though really Uparrow notation is only a recursive step away from a reasonable expression of g_64 (3^sup3^sup3... is stretching it a bit)
But fortunately TAN goes a good bit beyond that.

Anyways, what I want is a simple recursive definition for what happens if you add a tilde or arrow in front or on the back or whatever. (I kinda implied a definition but I have no idea how to express it.)


Also, what is the limit of TAN?
w^n for a finite n?
w^w?
Possibly w^^n for small, finite n?
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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 21st, 2019, 11:21 am

Would f(m) = the smallest n not expressible using m characters of mooseudocode be a computable or uncomputable function?
I'm guessing the latter since it's similar to rayo's function.
Is f(10^100) greater than or less than rayo's number?
What about f(10^101) ?
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Re: Thread for Non-CA Academic Questions

Post by toroidalet » July 21st, 2019, 3:54 pm

Moosey wrote:Would f(m) = the smallest n not expressible using m characters of mooseudocode be a computable or uncomputable function?
Uncomputable. It can directly simulate Minsky register machines in the obvious way (as well as a few other universal systems).
Moosey wrote:Is f(10^100) greater than or less than rayo's number?
f(10^100) is probably less than Rayo's number. Rayo's number is not the smallest number inexpressible by 10^100 symbols in first-order set theory, but the largest expressible number (plus one).




What is the constant k (if it exists) such that O(log(x!)) is between O(x^k) and O(x^(k-ε)) for some arbitrarily small ε? Does it depend on the base of the logarithm?

Is there a (different) constant k such that there are arbitrarily large integer powers differing by at most k?
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Re: Thread for Non-CA Academic Questions

Post by Gamedziner » July 21st, 2019, 5:28 pm

Moosey wrote:Would f(m) = the smallest n not expressible using m characters of mooseudocode be a computable or uncomputable function?
If "smallest" counts negative numbers, then, for all m, f(m)=-∞.

Code: Select all

x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!

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Re: Thread for Non-CA Academic Questions

Post by Moosey » July 21st, 2019, 5:35 pm

Gamedziner wrote:
Moosey wrote:Would f(m) = the smallest n not expressible using m characters of mooseudocode be a computable or uncomputable function?
If "smallest" counts negative numbers, then, for all m, f(m)=-∞.
That's not what I meant-- the smallest positive integer was what I meant.
Furthermore, -∞ is not a number.
toroidalet wrote:
Moosey wrote:Would f(m) = the smallest n not expressible using m characters of mooseudocode be a computable or uncomputable function?
Uncomputable. It can directly simulate Minsky register machines in the obvious way (as well as a few other universal systems).
Moosey wrote:Is f(10^100) greater than or less than rayo's number?
f(10^100) is probably less than Rayo's number. Rayo's number is not the smallest number inexpressible by 10^100 symbols in first-order set theory, but the largest expressible number (plus one).
Nice, so Mooseudocode is TC? Pretty good considering I came up with it in 15 minutes.


Unrelated question:
If I were to say,

Code: Select all

"I sometimes lie" I sometimes lie
would that be paradoxical? Or would it imply I am a habitual liar? Or is it possible I would only sometimes lie?
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Re: Thread for Non-CA Academic Questions

Post by dvgrn » July 21st, 2019, 8:00 pm

Moosey wrote:If I were to say,

Code: Select all

"I sometimes lie" I sometimes lie
would that be paradoxical? Or would it imply I am a habitual liar? Or is it possible I would only sometimes lie?
It's almost a paradox, if you read "I sometimes lie" to mean "there are times that I lie, and times that I don't lie". If that statement is a lie, then either you always lie, or you never lie. But it can't be that you never lie, because you're saying that you lie sometimes, and you'd have to be lying about that, but according to the premise you never do that.

So we'll have to go with the other option, which is that your statements are always lies. This seems pretty unlikely, but it's not a stand-alone paradox. The implication is that ' "I sometimes lie" I sometimes lie' is then also a lie. Seems to me there are two non-paradoxical ways out.

The main line of argument is that, again, we're saying that "sometimes" implies also that sometimes you don't lie. In that case, since we've figured out your statements must always be lies, you're technically lying by saying that you only lie sometimes. Or a bit less satisfying, more like an annoying technicality, you might not ever actually have said "I sometimes lie", so then you'd be lying in saying that you sometimes say that.

If "I sometimes lie" only means "It is possible to find a statement I have made that is false", without any implication that it is also possible to find a statement of yours that is true... then I think the only way to weasel out of paradox is on a technicality, as above.

Otherwise, if you're lying when you say that a false statement of yours can be found, then that implies that in fact you must not ever make false statements. But "I sometimes lie" is then a false statement. You may not ever have actually made that statement, but if you did, it was a lie -- and if you didn't, then you lied when you said that you made the statement! That seems to cover all the cases, so we have an unavoidable contradiction, and we can begin our merry ride round and round and down and down to the nonexistent bottom of the Whirlpool of Paradox.

EDIT: if the above isn't already way too much of this kind of thing for you, then -- assuming you've already run across what Douglas Hofstadter has to say about this topic -- I'd recommend getting hold of some of Raymond Smullyan's puzzle books, like The Lady and the Tiger. Or maybe Forever Undecided or The Gödelian Puzzle Book or a few others, if you happen to like pathological liar/truth-teller logic puzzles that gradually sneakily introduce you to Gödel's Incompleteness Theorem.

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