Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity.Cantor did claim actual infinity exists: — Devans99
We can prescribe how you would logically go about constructing the set of all natural numbers, but we cannot actually carry that process out to its completion.How exactly can the set of naturals be potentially infinite? — Devans99
Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality.It is defined as an actual infinity (all sets are actual). — Devans99
Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual? — aletheist
Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity. — aletheist
Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality. — aletheist
In my view, nothing within mathematics is actual--again, it is the science that reasons necessarily about strictly hypothetical states of affairs--and truth has to do with what is real, not just what is actual. Specifically, a proposition is true if and only if its subjects denote real objects and its predicate signifies a real relation among those objects. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while the actual is that which acts on and reacts with other things.Some parts of math are obviously true, or actual. — fishfry
We agree on this--the number 5 and the character of being prime are real, even though they are not actual.But there are truths that aren't physical. "5 is prime" is one of them. — fishfry
How so? Given my definitions above, perhaps what you mean is that "5 is prime" is true in the real world; in which case, again, we agree on this.But "5 is prime" IS true in the actual world. — fishfry
Okay, but you and I agree that this now-standard mathematical definition of a continuum is philosophically faulty; in my case, because I hold that a line is not composed of an actually infinite set of points. Instead, the continuous whole is ontologically prior to any discrete parts, which only become actual when we arbitrarily mark off a finite quantity of them for some purpose.What we are all taught at school - the Dedekind-Cantor continuum - a line is an actual infinite set of points - is an actual infinity. — Devans99
Please provide an authoritative source for this claim, or just acknowledge that you made it up. Here is what "well-defined" means in this specific context.Something that has potential but not actual existence is not well defined. — Devans99
Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80.It says there exists such a set — Devans99
In my view, nothing within mathematics is actual — aletheist
I am largely employing the terminology and philosophy of Charles Sanders Peirce in all this, so I will offer a couple of his examples. If I were to hold a stone in my hand and then release it, then it would fall to the floor. This subjunctive conditional proposition represents a real law--one that is true regardless of whether I ever actually carry out the experiment that it describes. Similarly, any diamond really possesses the character of hardness, regardless of whether anyone ever actually scratches it with corundum to demonstrate that it does.Again I confess I don't know what you mean by real versus actual. Can you give an example? — fishfry
When it is changed, it is not changed — Devans99
Imagine a hotel with infinite rooms and with an infinite number of guests a1, a2, a3,... Now imagine a new guest b1 who wants to stay in that hotel. The manager simply moves a1 to a2, a2 to a3, aN to aN+1,... and b1 gets a1's room. Notice that though the quantity, infinity is still infinity, hasn't changed, the quality has: b1 is the new guest. — TheMadFool
Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80. — aletheist
By that reasoning, an infinite set could have logical existence if our universe was infinite. But logical possibility is not at all dependent on actuality, or even metaphysical possibility. So infinite sets do exist, in the strictly mathematical sense of existence.A googol-sized set could have logical existence if our universe was bigger. — Devans99
It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets.A greater than any number sized set does not even have logical existence (leads to absurdities so it cannot be logically sound). — Devans99
It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets. — aletheist
Examples of abstract reality include the paradox of time, mathematics (Pi), cosmology (infinite universe theories), metaphysical phenomena, consciousness, so on and so forth.. — 3017amen
I wonder if it is simply an " illogical abstract " that actually exists in reality. Much like the metaphysical phenomenology of how the subconscious and conscious mind work together in an illogical manner (violating the laws of bivalence/LEM). — 3017amen
Beyond the boundary is nothingness IMO. Nothing cannot be actually infinite because it is nothing. If it is other universes then they cannot be actually infinite because it would lead to the absurdities referenced in the OP. Or see here for another example of the absurdity of actual infinity:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox — Devans99
You physically can't keep putting 10 balls in a vase, while only removing one. It's an unrealistic thought experiment. — Harry Hindu
If causation (space-time) isn't infinite, then you have to come up with an explanation as to how something came from nothing. — Harry Hindu
On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value.I have sometimes wondered about the length of 'now' - it seems it cannot be zero length else 'now' would be nothing (nothing length zero has existence). This type of thinking leads to consideration that time maybe discrete and eternal, over which the is much controversy. — Devans99
On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value. — aletheist
In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer.Discrete time would consist of discrete, non-zero, non-infinitesimal time slices - so they would have a duration. — Devans99
The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals.On the other hand, if time was composed of infinitesimal time slices, then each fixed period of time would be composed of 1/∞=0 length time segments giving a zero total length for all elapsed intervals. Which cannot be right. — Devans99
In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer. — aletheist
The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals. — aletheist
That is exactly what I described as discrete time, not what you described. Each individual frame of such a movie corresponds to an instantaneous state, with zero duration, arranged at a fixed and finite interval of 1/60th of a second. There is no flow of time between the frames, just a leap from one to the next.They are distinct, like a movie plays at 60 frames a second, each frame a time slice. There is nothing continuous about that. — Devans99
Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. The present moment includes an infinitesimal portion of the past and an infinitesimal portion of the future, which is why time does flow from one moment to the next.So infinitesimals cannot be the constituents of time because all time intervals would have zero length and time would not flow from one moment to the next. — Devans99
Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. — aletheist
With every circle,and with every other thing which has anything to do with π. With every measuring device with π marked on it (not too many of those). You're not distinguishing between number and numeral, which is typical since that distinction was made explicit to you by one the mathematicians writing here.I see that Pi cannot, in our reality... — Devans99
There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/∞. Again, if you truly want to understand, please read one or both of the short articles that I linked. If you prefer to remain ignorant, carry on.Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. As demonstrated in the op, ∞ leads to logical absurdities, so logically 1/∞ must be absurd too. — Devans99
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