Define Infinity - and is anything infinite?

Define Infinity - and is anything infinite?
Transcendantreason
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#31 - Quote - Permalink
Posted Jun 1, 2009 - 8:17 PM:

Further more it depends on how you define real or factual because in a sense your physical world is an active manifestation of your mind regardless of what philosophical opinions you harbor; so your question depends heavily on how you define your terms.
Transcendantreason
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#32 - Quote - Permalink
Posted Jun 1, 2009 - 8:33 PM:

I would define infinite as a never ending cycle of possible combinations, boundless, and immeasurable.
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#33 - Quote - Permalink
Posted Jun 1, 2009 - 8:50 PM:

Transcendantreason wrote:
I would define infinite as a never ending cycle of possible combinations, boundless, and immeasurable.

You go for it. The rest of us will stick to it's usual meaning. disapproval
Transcendantreason
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#34 - Quote - Permalink
Posted Jun 1, 2009 - 9:35 PM:

Hmm you mean to say that is not what it means...I would beg to differ; does my definition of infinity not serve substantial elaborate please. My definition of infinity is a manifestable archetype of the universal that is infiniteness, you mean to say it is in valid? Is what I said all together different from what any of you said... are there not idiosyncrasies between our our definitions? So is my idea wrong or merely an absraction of the concept?
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#35 - Quote - Permalink
Posted Jun 1, 2009 - 10:01 PM:

Transcendantreason wrote:
Hmm you mean to say that is not what it means...I would beg to differ; does my definition of infinity not serve substantial elaborate please. My definition of infinity is a manifestable archetype of the universal that is infiniteness, you mean to say it is in valid? Is what I said all together different from what any of you said... are there not idiosyncrasies between our our definitions? So is my idea wrong or merely an absraction of the concept?

I've been asked not to bite newbies, so I will have to word this with great care.

You said: "I would define infinite as a never ending cycle of possible combinations, boundless, and immeasurable'

Yet, plainly the cardinal numbers do not cycle. Are you going to maintain that they are not infinite?

And, the set of real numbers between 1 and 2 is bound; are you going to say it is also not infinite?

And, the set of real numbers between 1 and 2 is bigger (has a greater cardinality) than the set of cardinals; are you going to say that despite this it cannot be measured?

I'll stop there. Teeth in mouth, not in newbie.
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#36 - Quote - Permalink
Posted Jun 1, 2009 - 11:07 PM:

Something else to consider. The Koch snowflake has infinite length, yet fits in a circle.



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#37 - Quote - Permalink
Posted Jun 7, 2009 - 12:37 PM:

jehu wrote:
How about we simple say that some things can be finished (finite) and others cannot be finished (infinite)?
Far more better term than "finished/not finished" would be "measurable/unmeasurable". There lies the essence of mathematics.

Immeasurability is one of the main causes why we have a problem with infinity in mathematics. Just look from what mathematics developed: from a need to count things. Counting or something being computable is in the focal point of mathematics. If there is something that isn't computable, oh boy, then we have truly difficult problems in understanding the underlying logic. What is the reason for an uncomputable number to exist? Thats why many, even here, dismiss the idea of actual or absolute infinity and use only the Aristotelean notion of potential infinity.

Why? Well, simply because they think that because "there cannot be a greatest number because n < n+1", there cannot be an actual infinity. But this is nonsense: nobody is saying that infinity would be a finite number or behave in a similar fashion as finite numbers. It clearly cannot, so why on earth should we apply the logic of finite numbers to it? The obvious thing is that one simply cannot count to infinity: applying on number more to an finite number is still a finite number.

Our notion of a number is far too limited. That's why we have problems of understanding infinity.
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#38 - Quote - Permalink
Posted Jun 7, 2009 - 12:56 PM:

Clearly, that which cannot be finished necessarily cannot be measured, but it does not follow from this that the two terms are identical, for the same may be said of that which is unbounded, and we have already demonstrated that the terms “infinite” and “unbounded” are not identical.
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#39 - Quote - Permalink
Posted Jun 7, 2009 - 6:18 PM:

Very well said, Jehu.

And something that is unfinished, is not something precise. However, "the smallest" or "the biggest" seem to be something very precise."Unboundedness" is also something worth looking at. The opposite, something with clear bounds, is something one can hopefully measure.

Hopefully I'm not sounding as a broken record, but let's look at this problem from a different point of view: what kind mistakes have people done in math? Well, we make mistakes if we start with false premises. A well known mistake was to believe that "all numbers are rational numbers". It sounds totally reasonable. After all, if one thinks that math evolved from the need to count things, then natural numbers are obvious starting point. The to find ratios of numbers, or rational numbers, is logical also. And that is all.

But then it wasn't so.

Huge crisis. Denial. And the mistake? It was making a false assumption of what numbers can be. Now notice how this false assumption emerges. If we believe that "the math we use now holds all there is to know", then one can easily make such false assumptions (or presumptions). And thus one only stumbles into these things quite accidentally when doing math. And notice that again the problem with irrational numbers was a problem of measurement. And it didn't stop there: finally, the transcendental numbers came into view. Even if they can be very useful (as pi is), try measuring something exactly with those numbers. (And also remember how for instance in Wikipedia a number is defined: A number is a mathematical object used in counting and measuring.)

So what could numbers be more than just objects used in counting and measuring, as obviously not all numbers are usefull in these feats. In my view (and this isn't a definition of a number) numbers are something with an exact definition that separates each and every one distinctively from all the others. And this is my point: think about the infinitesimal or (absolute) infinity as a number with this broad definition: Can these be counted from other numbers? No. Somehow be measured? No. Are they distinctively definable from all other numbers? Yes. Do we stumble sometimes into them when doing math? Yes.

Hence the definition of the infinity is the real question.
jorndoe
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#40 - Quote - Permalink
Posted Jun 7, 2009 - 10:32 PM:

ssu (#39) wrote:

Hence the definition of the infinity is the real question.

So, what is the mentioned "absolute infinity" anyway?

By the way, WastingTime has a pretty good round-up of some of the topics over here:
Logic and Philosophy of Math » Does infinity exist?
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