Take the semantic of ‘21’ and the semantic of ‘divide’ and the semantic of ‘3’ and the semantic of ‘equals’. What does 21 divided by 3 equal? The answer is not 21/3 because that is the question that is being asked. It is not the answer. The answer is the semantic of 7. 7 is the answer in its actual purest form.
Now, what is 1 divide by 3 equal to? The answer is either absurd or .333… (where … stands for an infinity of 3s)
You cannot say the decimal system of listing numbers has issues. It is the semantic of ‘one’ and the semantic of ‘three’ and the semantic of ‘divided by’ such that when you ask what is one divide by 3? the answer is either you cannot divide 1 by 3 or it is zero point followed by an infinity of 3s.
Yes but semantics dictate that either 1/3 is absurd, or 1/3 is 0.333…
It is semantically inconsistent to say that 1 divide by 3 is 0.333… (follow by a vary large but finite number of 3s).
I see them as having the same identical answer. The semantic of 7. I don’t see them as being semantically identical. As in 21/3 is not semantically identical to 7/1. Their answers are semantically identical. This does not make them semantically identical.
It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as “semantics”, “infinite”, “…”, and “infinity”.
I don’t have any doubt that this whole thread is nothing but arguments over words and symbols.
Semantic = meaning
Infinite = that which has no beginning and no end
Infinity = that which no greater quantity can be conceived of
If I ask the question what is the answer to a third of 21 exactly, the answer I would get is 7. Not 21/3. Perhaps someone might say you will find the answer when you divide 70 by 10, but they are still saying you will find the answer after you do this division.
To convey what I’m trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that’s just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it’s .333…m or 33.333…cm
But that doesn’t mean that you can’t divide the meter. It only means that you cannot say it using decimal digits.
Bend that meter into a circle an cut it at exactly 120 degrees and you have your 1/3 meter. Ask someone how long those portions are in centimeters and they cannot give you an exact number.
I think I already did. (10^{\infty}) stands for “10 multiplied by itself infinitely”. Every term in that sentence (such as 10, multiplication and infinity) has an established definition. My claim is that all you have to do is use the established rules and deduce the meaning of the expression (in the same way one can deduce the meaning of “2 + 2” without having to look for an answer from mathematicians, books and online encyclopedias.) If you want a longer description, here is one: the statement tells you to calculate the result of (10 \times 10) and then take that result and multiply it by (10) to get a new result and then take that result and multiply it by (10) to get a new result and so on for an infinite number of times. What you get after you’re done multiplying all the (10)s is what the expression stands for. What exactly is unclear and/or unconventional about that? How can I help you if you are not willing to explain what’s unclear and/or unconventional? Merely repeating yourself by saying “It’s undefined”, “You haven’t defined it!” and “Define it!” will get you nowhere. A better approach would be to explain what it means for something to be undefined, and if it proves to be necessary, to present an argument that my expression is undefined.
Perhaps you should start by defining what the word “ambiguous” means.
When we say that what someone is saying is ambiguous, we are merely saying that we cannot determine exactly what that person is talking about (“He might be talking about X, but also, he might be talking about Y”.)
Wikipedia seems to be unambiguous about the meaning of the word “infinity”. It tells us right at the start that the word means “something that is larger than any real or natural number”. What exactly is ambiguous about that?
The fact that the word “infinity” can be used to represent more than one number does not mean that it is ambiguous. It merely means that it is not a specific number but a category of numbers (similar to the word “integer”.)
And even if “infinity” is an ambiguous term, what makes you think that logic can’t handle ambiguity? Of course, ambiguous terms have limited use but that’s a different subject.
The word “integer” cannot be used - as an integer - because it is mathematically ambiguous - it is a category of values, not a specific value. And “infinity” is equally a category of values - not a specific value.
(10^{\infty} = ?)
If you want to use maths and be specific just use (infA) instead -
(10^{infA} = 10^{Natural Number Count} = 10^1x10^2x10^3x…)
The only definition I could find for the ellipsis “…” is in Wikipedia -
I have never seen “…” being used to mean anything other than “continue indexed throughout the natural numbers”. So without a formal definition in our debate I would have stated - Ellipsis - “…” = “repeated throughout the natural numbers with index starting at either the first digit to the right of the decimal (when a decimal is to its left) or repeated throughout the natural numbers with index starting at the first left hand digit indicated at the left of the symbol.”
From there, most of our argumentation would never have occurred - although perhaps you want to debate a “better” definition - but that is a different debate.
Where .333… is meaningful, it is hypothetically possible to divide the meter into 3 equal parts. But the equality of those parts must be expressed by saying that each part is .333…m long. .333…m long is either meaningful (therefore it has an exact meaning, which means it can serve as an exact answer), or it is not meaningful (which means we cannot divide the meter into three meaningfully equal parts). It would be like saying:
A: I’ve just divided x into 3 equal parts.
B: How big is each part
A: Unknown in terms of exact size
B: How do you meaningfully know they are exactly equal?
or
A: I’ve just divided x into 3 equal parts.
B: How big is each part
A: It is exactly 1/3 of the whole of x
B: How big is 1/3 of the whole of x
A: I can’t give an exact answer to this
B: Does an exact answer actually exist but you just don’t know it, or is it the case that no exact answer exists?
A: No exact answer exists
B: You are saying 1/3 of x is meaningful, yet you are also saying that there is no objective truth regarding what 1/3 of x actually amounts to in terms of size/measure. How can one of those three pieces of x not have a measurable value? If it has a size, it has a measurable value.
But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.