The interval would be made so small that the length of the line was the same size as the width of the line - both being now effectively zero. — apokrisis
A line in turn would be arrived at as the constraint on the quality of “plane-ness”. Squish the 2D plane from either side and the limit of its compression becomes how a 1D line arises. — apokrisis
So the ontology is fundamentally complex. And hence not widely understood by folk....But good luck applying this kind of advanced systems logic to the simplicities of number theory. — apokrisis
↪keystone
Some of your quote links are not going to the posts in which the quotes occur. — TonesInDeepFreeze
If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element.
— keystone
You answered the question. You're not serious. — TonesInDeepFreeze
I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist. — TonesInDeepFreeze
And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way. — TonesInDeepFreeze
The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. — TonesInDeepFreeze
Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly.
— keystone
What on earth are you talking about? — TonesInDeepFreeze
I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it? — TonesInDeepFreeze
You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see? — TonesInDeepFreeze
Points are not "nothingness". — TonesInDeepFreeze
You're adding things into what I wrote that are not there. — TonesInDeepFreeze
Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? — keystone
I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.
— TonesInDeepFreeze
It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points. — keystone
the wooden stick upon which tic marks are placed is the continuum. — keystone
I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. — keystone
Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly.
— keystone
What on earth are you talking about?
— TonesInDeepFreeze
I'm referring to the objects of set theory being beyond our grasp. — keystone
The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences. — TonesInDeepFreeze
I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?
— TonesInDeepFreeze
I get what you're saying, but I don't agree with it. — TonesInDeepFreeze
You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see?
— TonesInDeepFreeze
Sometimes there's not enough room for two conflicting ideas. — TonesInDeepFreeze
Points are not "nothingness".
— TonesInDeepFreeze
It occupies zero space. — TonesInDeepFreeze
You're adding things into what I wrote that are not there.
— TonesInDeepFreeze
I'm confused why you embedded a geometric series into the definition. — TonesInDeepFreeze
We both think the other is not listening or being reasonable. — keystone
ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty". — TonesInDeepFreeze
So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on. — TonesInDeepFreeze
I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone
Wrong. Look up the math sometime. — TonesInDeepFreeze
Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze
Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness. — TonesInDeepFreeze
I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse. — TonesInDeepFreeze
Let N = the set of natural numbers.
Let f be a function.
Let dom(f) = N
Let for all n in dom(f), f(n) = 1/(2^n)
So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...
0 is not in ran(f).
Let g be a function.
Let dom(g) = ran(f)
Let ran(g) = {"off", "on"}
Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)
So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ... — TonesInDeepFreeze
ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".
— TonesInDeepFreeze
In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal. — keystone
So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.
— TonesInDeepFreeze
I dispute a continuum composed of points. — keystone
I take it that you consider the points to equal the continuum. — keystone
I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone
Wrong. Look up the math sometime.
— TonesInDeepFreeze
I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that — keystone
Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze
Then maybe ZFC is inadequate for analysis. — keystone
Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.
— TonesInDeepFreeze
I see points as emergent from distance — keystone
I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.
— TonesInDeepFreeze
Care to try? — keystone
Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, — keystone
Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?
Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc. — keystone
Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:
Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.
Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format? — keystone
And often we say that the naturals are defined as nested sets of sets. — keystone
I just can't envision any computer holding even just the natural numbers without exploding. — keystone
I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane — TonesInDeepFreeze
where we need to assume that the real line is composed of infinite points — keystone
You asked me about finitely many points, not about potentially infinitely many points. Be clear. — TonesInDeepFreeze
Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze
Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. — TonesInDeepFreeze
Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one. — TonesInDeepFreeze
I did not debate the definition of 'set'. — TonesInDeepFreeze
That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong. — TonesInDeepFreeze
We can add whatever math you want to my writeup...And still my point about the writeup stands. We have an infinite sequence. — TonesInDeepFreeze
Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences. — TonesInDeepFreeze
Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. — TonesInDeepFreeze
The description is not coherent, since it posits that there is a last state for a process that does not have a last state. — TonesInDeepFreeze
Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step. — TonesInDeepFreeze
It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask. — TonesInDeepFreeze
But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers. — TonesInDeepFreeze
if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite? — keystone
nested sets of sets containing no objects — keystone
If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you? — keystone
Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2? — keystone
I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. — keystone
Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm. — keystone
I'm not in a position to argue that Peano systems are inconsistent — keystone
Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.
And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.
I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane. — TonesInDeepFreeze
also cannot say that 1+1/2+1/4+1/8+... = 2 — keystone
There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". — TonesInDeepFreeze
'continuous' is defined in mathematics. I don't know what you mean by it. — TonesInDeepFreeze
There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.
Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too. — TonesInDeepFreeze
Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze
Ok, I get what you're saying now. — keystone
In that context, I don't mean 'system' in the sense of axioms and theory. I mean it in the sense of a tuple of a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply. — TonesInDeepFreeze
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