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See https://en.m.wikipedia.org/wiki/Hyperreal_numberDoes that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
I understand about every third word, but it does look interesting.
See https://en.m.wikipedia.org/wiki/Hyperreal_numberDoes that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
When I was a graduate student I had to mark undergraduate homework. One question asked for an example of an unbounded sequence. The answer given was:As I mentioned above, existence depends on what you're talking about. In the real number system infinite numbers may not exist but they certainly do exist in a hyperreal number system and do obey the normal rules of addition and multiplication. In that case, an infinite plus one is infinite.
It is "not finite", but ##w+1 \neq \omega## - they are different hyperreal numbers.As I mentioned above, existence depends on what you're talking about. In the real number system infinite numbers may not exist but they certainly do exist in a hyperreal number system and do obey the normal rules of addition and multiplication. In that case, an infinite plus one is infinite.
That is not a sequence of real numbers, and if the question is not about real numbers, it is unclear what "unbound" means. In the hyperreal numbers, and assuming ##\infty## means ##\omega##, the sequence is bound (e.g. by ##\omega##).When I was a graduate student I had to mark undergraduate homework. One question asked for an example of an unbounded sequence. The answer given was:
##1, 2, \infty, 4, 5 \dots##
How would you assess that?
No, I think we said essentially the same thing. He did say infinity plus one equals infinity but there is no number called infinity, there are infinite numbers. So, technically, an infinite number plus one is another infinite number. If you're interested, Jerome Keisler has made his intro calculus text available for free. Sections 1.5 and 1.6 contain a discussion of the hyperreals.Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
I couldn't disagree more.Invoking the hyperreals in a "B" level thread is the maths equivalent of invoking the stress-energy tensor to explain the SHM of a pendulum!
The hyperreals are at an advanced undergraduate level and depend on a solid grasp of real analysis. They are not suitable for a "B" level thread, IMHO.
I never said they were the same, I said they were both infinite.It is "not finite", but w+1≠ωw+1 \neq \omega - they are different hyperreal numbers.
A Basic level thread is one that is suitable for High School students and assumes the appropriate knowledge. Hyperreals do not come into that category.I couldn't disagree more.
Is it safe for me to assume that you were simply speaking from the "real number" point of view in this answer?Infinity + 1 is undefined
He actually said the opposite...He did say infinity plus one equals infinity
It is "not finite", but w+1≠ωw+1≠ωw+1 \neq \omega - they are different hyperreal numbers.
Again, I disagree. Hyperreals have been used consistently and successfully in teaching calculus to high school students for decades. I'm not sure what your objection is. They are also taught irrationals and use and understand them successfully without any thought to the construction of the reals which most of them will never see.A Basic level thread is one that is suitable for High School students and assumes the appropriate knowledge. Hyperreals do not come into that category.
We have rules on PF because it's a serious science and maths forum. You need to adhere to these rules.
I'm confused here & may be misunderstanding; but I would have assumed that we wouldn't want to conflate the human act of counting (i.e. the natural numbers, or any other number system we invent) with things that exist in nature - if that's what you mean by "really exist"? (Though maybe you mean something else?)There are different levels of "infinity" which are represented by cardinal numbers, ##\aleph_0, \aleph_1, \aleph_2, ...## . . . I would say that they are not imaginary. They measure things that really exist, just like natural numbers do.
We do not say that 1/infinity = 0. At least not in the field of real numbers (or hyper-reals for that matter).And while infinity + 1 = infinity indicates it is not truly a number, we say that 1/infinity = 0, so there we do use it as a number.
OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?We do not say that 1/infinity = 0.
Are you thinking of infinity in a sense of (n/0) where n is any real number, and thus...OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?
There is a distinction between a limit and a quotient, yes. There is no rule that says that the limit of the quotient must be equal to the quotient of the limits when one of the two fails to exist.OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?
So my question is, still, how much is 1/infinity?There is a distinction between a limit and a quotient, yes. There is no rule that says that the limit of the quotient must be equal to the quotient of the limits when one of the two fails to exist.
Since "infinity" is not a number, 1/infinity is not defined.So my question is, still, how much is 1/infinity?
In that case I, and many other people, have done many problems incorrectly.Since "infinity" is not a number, 1/infinity is not defined.
You need to understand something basic about mathematics. Mathematics is (by and large) about careful definitions, precise axioms and their logical consequences. There are lots of possible sets of definitions and no one true "right" set.In that case I, and many other people, have done many problems incorrectly.
I like the above post, including the qualification I have quoted here. This agrees entirely with what I have been reading lately in a few different books, including the Tim Gowers book I mentioned, https://www.amazon.com/dp/0192853619/?tag=pfamazon01-20. Gowers, by the way, is a prof at U. of Cambridge in the U.K.. who seems pretty highly credentialed; all I know about him is that I like his writing.You need to understand something basic about mathematics. Mathematics is (by and large) about careful definitions, precise axioms and their logical consequences. There are lots of possible sets of definitions and no one true "right" set.