Re: The infintely small number b
- From: BuddhaThu <softspokenbuddha@xxxxxxxxx>
- Date: Sat, 17 Nov 2007 12:48:06 -0800 (PST)
Let me see if I can put it more clearly.
An infinity that is set by a boundary must obey the boundary rules. To
merge 1 to an infinite set will be a violation of the boundary rules.
They must be kept in separate cardinal areas when performing
arithmetic operations.
This means 'summing' has a distinct flavor than ordinary finite
arithmetic.
Transfinite arithmetic is different from ordinary finite arithmetic.
The behavior is similar, but not the same.
Sorry for the spelling errors. I forgot to spell check. -- B.T.
-----------
Let me take a crack at this.
When dealing with the continuum, there is need to see that we are
dealing with two distinct structures. The first is the infinite real
number line and the second is the infinitesimal number line.
Their distinction is as follows.
The real number line no matter how small will always be finite.
But once you can generate a rule, you can have the infinitesimal.
The infinitesimal is a generalization of quantity. You cannot sum
infinity. You cannot subtract infinity. You cannot multiply or divide
infinity. It simply does not make sense. But you can generalized
infinity into a mathematical rule.
In this, I am with Gauss on relating to any infinite structure.
Infinity is not a number, but a quality of things. To say how infinity
adds or subtracts is to say how beauty adds or subtract. It does not
make sense, and in a way, is a wrong grammar rule. It is not something
that can be dictated with quantity, but of quality.
However, once you can generalize it, as in distinct from summing it,
into a set, then this generalized set with its own set of rule
governing boundaries can exert a fair amount of addition, subtraction,
multiplication, and division. It is now a quantity. It is now
countable. This is because it is behaving as if it is finite. Finitude
here is within a boundary set by a rule.
In this instance, you can order them into ordinals and compare them.
You can split them, as long as they are reasonably defined within
their own finite set of boundaries.
When we say infinity + 1, we are not saying that 1 is absolutely
merged into that infinity. It cannot sum.
But we can say infinity + 1 within their own rigidly separated sets of
quantity. This makes them into two cardinals instead of one. With
this, you can set an ordering mechanism, much like an ordinal process.
This is if you keep them in their separate cardinal formulas. But once
you merge, then the whole deal is off.
Infinity + 1 will not make sense.
.
- References:
- The infintely small number b
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- Re: The infintely small number b
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- The infintely small number b
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