So I take this to mean that when we cannot determine the number of something (e.g. how many people there are in the universe) that something (e.g. the total number of people in the universe) is said to be unquantifiable. In other words, to say that something is unquantifiable is to say that we can’t measure it (because we don’t have the means to do so.)
However, when we say that the number of something is infinite, we are specifying that something’s quantity, we are NOT saying that that something is immeasurable.
Allow me to circumnavigate the linguistic nightmare…
How many apples do I have if I had infinite apples and you removed one?
What is the NEW quantity that is produced?
Because if there is no NEW quantity, if I still have infinite apples… it’s exactly the same difference when I take an infinitesimal fraction away from the number 1…
You have a smaller infinity of apples. Infinities come in sizes (something Ecmandu denies), so there are bigger and smaller infinities.
It’s basic logic. You have an infinite number of apples. You remove one from it. How many apples are you left with? You can’t say you have the same number of apples as before because that would contradict your statement that you removed an apple. (Unless, of course, a new one appeared elsewhere, but no such thing was stated.)
The word “infinity” is typically taken to be a reference to a category of numbers. If you have a number belonging to this category of numbers and you subtract one from it, the result will also be a number belonging to that category of numbers. Thus, if you have an infinite number of apples (in the sense that the number of apples is greater than every integer) and you subtract one from it, you’d still have an infinite number of apples (in the sense that the resulting number would still be a number greater than every integer.) But that does not mean the two numbers are equal. It merely means they are the same type of number (i.e. infinite.)
That would be a contradiction. You said that you subtracted from infinity i.e. that you took something away from it. You cannot now say that you took nothing away from it.
Because you believe in ‘orders of infinity’ you see infinity as an OBJECT and not a PROCESS!!!
This is a very important distinction!
The number 1 is an object (as far as mathematical entities are objects) - you can do things to mathematical objects and they will change.
Infinity is not an object however! Infinity is a process, this is the same reason omniscient beings are false, omniscience is a process, not an object!
Nobody can hold infinity ‘in their head’, the act of trying this just leaves you with something that keeps expanding! There’s no way to measure an infinite process is greater than another infinite process, because you never get to the end to say, “wow, I’m at the end of this one and it’s bigger than the other one!
Let me give you a concrete example!
1,2,3,4,5,6,7…
2,3,4,5,6,7,8…
Still 1/1 correspondence!!!
It’s almost analogous to the dual slit experiment to this regard … when you try to calculate it, the act of observation changes it.
Some infinite sets are larger than other infinite sets
Some infinite sets are smaller than other infinite sets
How do we determine this ? By simply calculating the size of an infinite set relative to any other infinite set and so for example
The infinite set of integers is greater than the infinite set of primes because integers occur more frequently on the number line
And so it is frequency that determines the size of an infinite set but one does not have to count all of the members of one to know this
As that would be an impossible task even a computer at the speed of light could not calculate as it would require infinite time to do so
But different size infinities have been known since Cantor first discovered them
These two contradict each other. As long as you can make correspondence, there are no orders of infinity!
Cantor thought (incorrectly) that there are “uncountable infinities”
Because of this mistake, he thought that some infinities are greater than others!
Even cantor himself would agree that your answer is false… the correspondence between counting numbers and primes is obvious and not an example of orders of infinity.
Aleph null is the countable set of all natural numbers but every infinite set greater than
it is uncountable because it can not be mapped one to one as its cardinality is too large
To say that you subtracted from infinity is to say that you took SOMETHING away from it. And if you say that you took SOMETHING away from infinity, you cannot then say that you took NOTHING away from it. That would be a logical contradiction i.e. P and not-P.
And no, I do not see infinity as an object. Infinity is neither an object nor a process. It’s a quantity.
The statement “An infinite number of apples” does not describe a process. It’s a description of the universe (some part of it) at a single point in time. If you’re not describing some kind of change (and that means the state of the universe at more than one point in time), you’re not describing a process.
There’s no 1:1 correspondence between the two sets. The first set is larger than the second. It logically follows from your claim that you produced the second set by making a copy of the first set and removing one element from it. To say that they are equal in size is to contradict that claim.
You are misusing words. Subtraction operates on QUANTITIES. It does NOT operate on objects and processes.
removing 1 from infinity implies you took something away… the problem is “infinite” is not a number, it’s not a quantity, it’s a concept of endlessness, that concept is not altered by removing a portion.
If there is endless space and we move ourselves 1 mile in any given direction we do not have “less space” in that direction as a consequence.
Infinite sets can be quantified RELATIVE to each other. For example a set of only even numbers is smaller than a set of all whole numbers, because for each entry into one set there are two entries in the other, but that’s neither here nor there.
I can see how it might be difficult to map it the other way around…
But imagine I told you I was going to reduce the length of a finite line by an infinite fraction of it’s length… that means no matter how close to the border of the line I get, I will never arrive at the portion that I WILL cut…
What I’m communicating in that concept is a scenario where the line NEVER gets reduced… 1 is never reduced if I reduce it by 0.<insert infinite 0’s > followed by a 1 at the end
0.999… is therefore equal to 1 as the difference between them is an endless string of 0.0000… asserting there is a 1 at the end of an endless string is semantic nonsense.
I’ve heard this argument before! If hyper-reals don’t exist (which I don’t think they do either) then there is no possible space between 0.999… and 1, thus, they must be equal as two sides of the same number; the wave and the photon (so to speak)
What I keep iterating is that if infinite series converge at limits, then, every real and imaginary number must equal zero!! As well as it’s general convergence, say, an infinite series equals 2!
This means that it both equals 2 and zero at the same time! That makes it undefined!
So I ask you this! Are all infinite series undefined? Or! Are they just what they appear to be; infinite series that never converge?!
To say that you removed (or subtracted) one thing from a group of things is to say that you ALTERED that group of things and that you altered it in a very specific way, namely, that you REDUCED the number of things the group consists of.
You can argue that an infinite number of apples does not refer to a group of things that can be altered (even though that’s not true) but in that case you cannot say that by removing an apple from that group of apples you do not alter that group of apples. That would be a logical contradiction. If you say that a group cannot be altered, you cannot then say that you subtracted something from it (since subtraction IMPLIES change.)
If you say that the number of an infinite number of apples cannot be altered, what follows is that you cannot subtract an apple from that group, it does NOT follow that by subtracting an apple from that group you get the same number of apples as beore.
And as I briefly mentioned, the idea that the statement “An infinite number of apples” refers to a group of apples the number of which cannot be altered is simply not true. When someone says “At the present moment, the total number of people in the universe is infinite” they are NOT saying that the number of people in the world cannot be changed (e.g. by every person in the universe dying thereby reducing the total number of humans to zero.)