The idea that there actually exists an infinitely large anything! would literally mean that whatever that thing is, it escapes (outreaches) such that it’s not even connected to its preceding chain of existence via continuity, i.e., a break in the chain of existence, and therefore cannot possibly exist.
But for supposed infinitesimal values the opposite happens, i.e., it gets squashed out of existence – it has to be that infinity refers to some value above 0 else it would literally have no meaning. But at the same time, it cannot be represented by a numerical value – any such value wouldn’t comply with infinity’s definition (i.e., equivalent to being too large, just by existing). Therefore, infinity eliminates itself from existence via its own definition and common sense.
The explanation given here is equivalent to the above section: “Infinity Gets Squashed vs Infinity Escapes”.
The implication of the term “direction” as used here is – increasing in absolute value vs decreasing in absolute value.
In contrast to the above, consider direction as used more typically:
It is perhaps more intuitive to realise what is meant by actualised when considering something to be increasing in value, e.g., distance in space extending (increasing in value direction), such that incomprehensible locations already exist (i.e., not having positional coordinates relative to our location), regardless of whether any physical matter has ever existed there – of course it’s impossible for such supposed locations to exist.
In contrast, infinitesimal values by definition of infinity (i.e., not the misnomer: “potential infinity”) are also bound to exist by the same logic – infinitely small values of things must already exist physically, or be truly represented theoretically, e.g., for physical objects, there must already exist infinitesimal physical structure (property of being physically divisible). Keep in mind! the action of dividing something indefinitely is not infinity any more than travelling indefinitely is infinity.
If you divide physical structure to the extent of considering just 1 point-like particle, then either it has no physical measurement (e.g., photons are massless; dimensionless – technically not occupying a position in 3D space), or it has a very small measurement – if so-called “point particles” do actually have size which can be further divided, then even smaller particles might one day be discovered.
The Planck length is approximately 10-35 metres, which is the same as 10-20 (1 / (10 * 10 * 10... to 20)) times smaller than the diameter of a proton. If a measurement smaller than the Planck length is ever theorised, it will not be infinitely small.
Another, perhaps more intuitive way to think about how infinite and infinitesimal are equivalent to each other, is to consider:
Simply one continuous sequence heading in opposite directions from the neutral position: “5”
This also means that from the perspective of any (supposed) infinitesimal existence, all finite values are infinite in size, i.e., the same size, but of course that makes no sense whatsoever as explained throughout this website – hence, infinity is a misconception.
Realistically of course, it seems unlikely that microscopic physical structure and the vast distance of space, just happen to exist being represented by a perfectly divisible numerical sequence.
A numerical example:
Hypothetical smallest physical structure: 1mm.
Divided result: 50, 25, 12.5, 6.25, 3.125, 1.5625, 0.78125
Yield discrepancy: 1 - 0.78125 = 0.21875
Hypothetical longest distance: 1000mm.
Expanded result: 50, 100, 200, 400, 800, 1600
Yield discrepancy: 1000 - 1600 = -600
Considering that a smallest distance does indeed exist – how does movement occur?
Answer: matter first exists at 1 position, then the next position, then the next, and so on.
If matter stops existing at its current position, where does it go until it exists at the next position?
Answer: massive-type particles (matter) interact with 4D spacetime itself – effectively popping in and out of spacetime's 3D projection of space, i.e., our perceived experience of reality.
Moreover, even if we disregard the “property of being physically divisible” as mentioned above, any supposed infinitesimal division of physical reality must still be represented by a numerical value above 0 to exist, and yet if that value is multiplied by infinity, the result is also infinity – i.e., it makes no difference whether considering objects (physical or platonic) or theoretical numerical values – no infinitesimal quantity exists.
A more detailed explanation about: Platonic numerical paradoxes – a link to the “Platonic Infinity “...” Numerically Paradoxical” page.
The paradox being identified here, is that infinity by definition doesn't allow existence by numerical value (is ridiculous), i.e., any location that can be referred to via coordinates relative to our location, or any division that can be referred to as having a numerical size, does not represent an infinite or infinitesimal existence, i.e., existence without a measurement.
The unequivocal conclusion via pure logic alone, is that if supposed infinitesimally small structures of matter must have no numerical size to qualify as being infinitesimal (i.e., by their definition), then they simply don’t exist, just like locations cannot exist that don’t have relative coordinates to ourselves.