But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry
That's exactly why the reals are regarded as a mathematical continuum, and the rationals aren't. — fishfry
We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful. — sandman
Therefore you can make a universe out of a pebble — Gregory
The conclusion is that the limits of calculus go out the window!
I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion — Gregory
If things are purely finite, there is a smallest unit of space. That is impossible though because you just divide it further. If you can't it's zero and has nothing to do with the object. — Gregory
This is a common misconception. What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself.I think modern science has demonstrated that there is a smallest unit of space. — Metaphysician Undercover
What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself. — aletheist
By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. — aletheist
This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves.What we call "space" and what we call "time", are abstract ideas created to account for what we observe. — Metaphysician Undercover
That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. As George Box famously put it, "All models are wrong, but some are useful.""Continuum" as implied by common usage means a collection of contiguous, but separate individual units. — Metaphysician Undercover
Only in the sense that continuity is a property, while a continuum is anything possessing that property."Continuum" and "continuity" have very different meanings. — Metaphysician Undercover
This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves. — aletheist
That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. — aletheist
Only in the sense that continuity is a property, while a continuum is anything possessing that property. — aletheist
If you adamantly deny the reality of space and time, then there is nothing more for us to discuss on that front.There's no such thing as real space and times. — Metaphysician Undercover
The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not.You can't say that the mathematical definition is wrong, because it's a mathematical term. — Metaphysician Undercover
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