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Cont: Deeper than primes - Continuation 2

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More generally, {y}~$y OR {y}$y simply means that {y} is not necessarily a successor of y.

By wff ,if R is an n-ary relation symbol, and t1,...,tn are terms, then R(t1,...,tn) is an atomic formula.

In the considered case R is $ where the terms are ({y},y) so we get the atomic formula $({y},y).

Also by wff, If φ and ψ are formulas, and • is any binary connective, then (φ • ψ) is a formula, and in our case (φ OR ψ) is a formula.

Also by wff, if φ is a formula, then ~φ is a formula, so ~$({y},y) is also a formula, and also $({y},y) OR ~$({y},y) is a formula.

$({y},y) can be written also as {y}$y and ~$({y},y) can be written also as {y}~$y, so we get {y}$y OR {y}~$y, which means that {y} is not necessarily a successor of y (as written above).
 
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Apology, I haven't read the whole thread, and it seems to have been going on for years. However I did see that there were questions concerning the number
0.999... and its equivalence to 1. Now, I might completely misinterpret the discussion but it seems to me that whenever such questions appear, the deeper
underlying problem always seems to be an improper definition of the continuum. This is where all of those paradoxes come from.
Again, don't know if this is relevant to this thread but I can provide my own ideas on this, if people are interested.
Let's start with Achilles and the Tortoise. Achilles is very fast, so the Tortoise is given a head start in their little competition.
Now, whenever Achilles moves, the Tortoise moves as well. Slower maybe, but he still moves. And since there's never a moment where Achilles moves and the
Tortoise doesn't advance his position, from this Zeno concluded that Achilles could never overtake the Tortoise. And since there's obviously no such
problem in the real world, you have a paradox. So what's wrong? Measure is improperly defined. Measure or lenght, area and volume, cannot be defined
by summing points on the real line. A line might be composed of an infinite number of points, you cannot define the length of a line in this way without
running into paradox. A proper measure function, that obeys the rules of summation, needs to be defined. Then the Achilles/Tortoise problem disappears.
There's also another way to remove the paradox using smooth infinitesimal analysis, where a point doesn't have zero measure, (which is the source of the
paradox, that a point has dimension zero,) but infinitesimal measure. And summing points as infinitesimals is an integral, obeying the rules of a proper
measure.
(For those of you interested in my Religious theory, you might want to check out Liber 418 by Aleister Crowley, The Cry of the 5th Aethyr.
The Vision of the Arrow therein described is a version of Zeno's Arrow Paradox, but whereas the paradox can be resolved by relaxing the law of excluded
middle (smooth infinitesimal analysis), the Vision describes a higher problem of the continuum to be resolved by relaxing the law of no contradiction.)
 
jsfisher said:
Not in anything even vaguely like Mathematics. Time (again) for you to review what constitutes a well-formed formula in propositional calculus.
Click to expand...
http://www.internationalskeptics.com/forums/showpost.php?p=11269502&postcount=1344, http://www.internationalskeptics.com/forums/showpost.php?p=11269567&postcount=1345.

In other words, you still have no argument about the last discussed subject.

jsfisher said:
{{A, B}} either is or it is not, pick one,
Click to expand...

{{A, B}} either is or it is not,
 
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TheAdversary said:
Apology, I haven't read the whole thread, and it seems to have been going on for years. However I did see that there were questions concerning the number
0.999... and its equivalence to 1. Now, I might completely misinterpret the discussion but it seems to me that whenever such questions appear, the deeper
underlying problem always seems to be an improper definition of the continuum. This is where all of those paradoxes come from.
Again, don't know if this is relevant to this thread but I can provide my own ideas on this, if people are interested.
Let's start with Achilles and the Tortoise. Achilles is very fast, so the Tortoise is given a head start in their little competition.
Now, whenever Achilles moves, the Tortoise moves as well. Slower maybe, but he still moves. And since there's never a moment where Achilles moves and the
Tortoise doesn't advance his position, from this Zeno concluded that Achilles could never overtake the Tortoise. And since there's obviously no such
problem in the real world, you have a paradox. So what's wrong? Measure is improperly defined. Measure or lenght, area and volume, cannot be defined
by summing points on the real line. A line might be composed of an infinite number of points, you cannot define the length of a line in this way without
running into paradox. A proper measure function, that obeys the rules of summation, needs to be defined. Then the Achilles/Tortoise problem disappears.
There's also another way to remove the paradox using smooth infinitesimal analysis, where a point doesn't have zero measure, (which is the source of the
paradox, that a point has dimension zero,) but infinitesimal measure. And summing points as infinitesimals is an integral, obeying the rules of a proper
measure.
(For those of you interested in my Religious theory, you might want to check out Liber 418 by Aleister Crowley, The Cry of the 5th Aethyr.
The Vision of the Arrow therein described is a version of Zeno's Arrow Paradox, but whereas the paradox can be resolved by relaxing the law of excluded
middle (smooth infinitesimal analysis), the Vision describes a higher problem of the continuum to be resolved by relaxing the law of no contradiction.)
Click to expand...
Please do not hijack threads.

You can open your own thread.
 
Sorry for another hijack attempt :) This is my last one :

The concept of infinity and it not being a number came up. That's right, notions of infinity have a nasty habit of running into paradox. That's why for
infinitesimals and their reciprocals we require an extended number line. Ever heard of non-standard analysis? It can deal with infinities, infinitesimals and
infinities can be larger or smaller than each other without contradiction.
I just think a lot of endless ( :) ) discussions on infinities stem from using the wrong number system.
Check out non-standard analysis. That's the first time in history the infinitesimals were given a proper definition.
But anyway, I'll remove myself from this thread. Bye!
 
Doron,

Back in post #1291, you confirmed that your IsSuccessorOf operator was this:
(1) X IsSuccessorOf Y iff X = {Y}​

Then in post #1309, you confirmed that your IsComplete operator was this:
(2) IsComplete X iff NOT({X} IsSuccessorOf X)​

Since then, you have also provided a great many non-sequiturs, contradictory, and confusing posts that have only served to muddy things for no purpose.

So, for the purposes of getting the signal-to-noise ratio well above zero, would you please either confirm statements (1) and (2) are correct or provide the appropriate corrections.
 
jsfisher said:
Doron,

Back in post #1291, you confirmed that your IsSuccessorOf operator was this:
(1) X IsSuccessorOf Y iff X = {Y}​

Then in post #1309, you confirmed that your IsComplete operator was this:
(2) IsComplete X iff NOT({X} IsSuccessorOf X)​

Since then, you have also provided a great many non-sequiturs, contradictory, and confusing posts that have only served to muddy things for no purpose.

So, for the purposes of getting the signal-to-noise ratio well above zero, would you please either confirm statements (1) and (2) are correct or provide the appropriate corrections.
Click to expand...
jsfisher,

Please do not force on me your iff , because I do not use it in the considered subject.

As long as you keep doing that, anything that I express is inevitably "non-sequiturs, contradictory, and confusing..." (including, for example, http://www.internationalskeptics.com/forums/showpost.php?p=11269619&postcount=1348).
 
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doronshadmi said:
jsfisher,

Please do not force on me your iff , because I do not use it in the considered subject.
Click to expand...


And yet that's what you previously confirmed. It was not until you found yourself trapped in contradiction that you have denied or attempted to re-qualify them.

Ok, so how would you like set completeness defined? (And, yes, since sets will need to be either complete or not, an if-and-only-if will be a required part of the definition.)
 
For the last time, by using wff {y}$y OR {y}~$y.

If {y}~$y then y is complete.

If {y}$y then y is incomplete.
 
doronshadmi said:
For the last time, by using wff {y}$y OR {y}~$y.

If {y}~$y then y is complete.

If {y}$y then y is incomplete.
Click to expand...


Equivalent to IsComplete X iff NOT({X} IsSuccessorOf X) under the definition for IsSuccessorOf you provided.

Any change needed to the IsSuccessorOf definition?
 
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jsfisher said:
Ok, so how would you like set completeness defined? (And, yes, since sets will need to be either complete or not, an if-and-only-if will be a required part of the definition.)
Click to expand...
The keyword here is OR, as follows:

(y is complete if {y}~$y) OR (y is incomplete if {y}$y)
 
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jsfisher said:
Your lack of facility with boolean algebra is quite incredible.
Click to expand...
This is not an informative reply.

iff is (A => B) AND (B => A).

if is (A => B).

Please use your mathematical skills in order to reply about:

1) (IsComplete X iff NOT({X} IsSuccessorOf X)) OR (IsInComplete X iff ({X} IsSuccessorOf X))

2) (y is complete if {y}~$y) OR (y is incomplete if {y}$y)
 
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doronshadmi said:
...snip for focus...
Click to expand...

The task at hand is to define the IsComplete operator. Showing us miscellaneous expressions the include the IsComplete operator does not define it.

You need to provide an expression that shows to what IsComplete S is equivalent. Something of the form:
IsComplete S <=> ...​
 
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