On 9/1/2021 9:36 AM, William wrote:
> On Wednesday, September 1, 2021 at 9:28:50 AM UTC-3, WM wrote:
>> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>
>>> A "gap" that changes is not capable of containing a real number
>> Take the interval [0, 1/n] for n in |N. It changes
>
> Piffle. If it changes it is not an interval. And what it contains is not a real number as a real number does not change.
>

WM is using n as a "variable", whatever that is

On 9/1/2021 6:26 AM, Greg Cunt wrote:
> On Tuesday, August 31, 2021 at 3:13:33 PM UTC+2, WM wrote:
>
>> like the largest known prime number
>
> "the largest known prime number" is mumbo-jumbo
>

from the Ant list 6.0

675 Mumbo Jumbo Ants with Dementia

On 9/1/2021 7:31 AM, Gus Gassmann wrote:
> On Wednesday, 1 September 2021 at 08:26:19 UTC-3, Greg Cunt wrote:
>> On Tuesday, August 31, 2021 at 3:13:33 PM UTC+2, WM wrote:
>>
>>> like the largest known prime number
>>
>> "the largest known prime number" is mumbo-jumbo
>
> I wouldn't be quite so harsh. Yes, there is a time dependence that should be made explicit, but we know (!) that the largest known prime at any moment is a Mersenne prime, and even with GIMPS there is a gap of at least three years (on average) until the next one is found. However, the rest of his contribution in that post *is* mumbo-jumbo.
>

na, it's all mumbo-jumbo, WM does not speak Math.

"the largest known prime number" is also red herring

On 9/1/2021 6:38 AM, Greg Cunt wrote:
> On Tuesday, August 31, 2021 at 10:47:26 PM UTC+2, WM wrote:
>> FromTheRafters schrieb am Dienstag, 31. August 2021 um 17:52:54 UTC+2:
>>>>
>>>> [...] there are dark numbers. They fill the gap and make the real line continuous.
>>>>
>>> We have that already, without a need for dark numbers.
>>>
>> You have <bla bla bla>
>
> It seems that you have lost you last marbles, Mückenheim.
>

there is no "last" marble

On Wednesday, September 1, 2021 at 2:31:25 PM UTC+2, Gus Gassmann wrote:
> On Wednesday, 1 September 2021 at 08:26:19 UTC-3, Greg Cunt wrote:
> > On Tuesday, August 31, 2021 at 3:13:33 PM UTC+2, WM wrote:
> > >
> > > like the largest known prime number
> > >
> > "the largest known prime number" is mumbo-jumbo
> >
> I wouldn't be quite so harsh. Yes, there is a time dependence that should be made explicit, but we know (!) that the largest known prime at any moment <<

See the difference? The very point (for me) is the following: /the largest known prime number in 2010/ is not /the largest known prime number in 2020/.. There is not some "changing" number "like the largest known prime number" (WM).

Right, in everyday language we often don't mention the time dependence explicitly. But in the context of logic and mathematic it's a MUST.

For example: "the oldest living man" => actual meaning: "the oldest living man _now_" or "the oldest living man _todate_" etc. From a later "perspective" we have to add the date of uttering/printing (etc.).

"the largest known prime number" (without ANY time reference) is simply nonsense.

For Mückenheim this expression is referring to a changing mathematical object. Go figure...

On Wednesday, September 1, 2021 at 2:28:50 PM UTC+2, WM wrote:

> Take the interval [0, 1/n] for n in IN. It changes and

No, THE interval [0, 1/n] for A CERTAIN n in IN does not change.

Actually, no interval [0, 1/n] (with n e IN) changes.

Of course two intervals [0, 1/n] and [0, 1/m] are DIFFERENT if n =/= m.

Holy shit...

Hint: If we are talking about "changing so-and-so" in math its only a "figure of speech". Mathematical objects don't (actually) "change".

On Wednesday, September 1, 2021 at 2:36:32 PM UTC+2, WM wrote:

> every finite set of endsegments [...] has an infinite intersection. The infinite set [of endsegments] has an empty intersection, but you cannot describe most of its endsegments.

Why do we have to "Mückenheim-describe" them?

In math it suffices to DEFINE the notion of an endsegment (and the set of all endsegments) to be able to PROVE certain things concernig endsegements (and sets of endsegments).

For example: We are able to PROVE: Each and every finite set of endsegments has an infinite intersection. Each and every infinite set of endsegments has an empty intersection. Period.

On 9/1/2021 8:28 AM, WM wrote:
> William schrieb
> am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:

>> A "gap" that changes is not capable of containing a real number
>
> Take the interval [0, 1/n] for n in |N. It changes and
> contains many real numbers for every n.

If you change what n refers to, you are equivocating.
Equivocating is a fallacy.

If you know that n is in N, then you also know many other
claims about n. Which individual n refers to is not one of them,
but many other claims.

If, later, you learn that (n-1)*(n-2) = 0, you still don't
know which individual n refers to. But you will know that
n is not outside of {2, 3}.

Take the interval [0, 1/n] for n in |N. It doesn't change,
but we know, even without knowing which element of N it is
that n refers to, that [0, 1/n] contains many real numbers.

We can know claims and reason about claims without knowing
which specific individual we are referring to, if we only
make claims that are true of any individual we might be
referring to.

But "changing" what we refer to is chaos. Will we have
n not equal to n?

On 9/1/2021 11:58 AM, Jim Burns wrote:
> On 9/1/2021 8:28 AM, WM wrote:
>> William schrieb
>> am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>
>>> A "gap" that changes is not capable of containing a real number
>>
>> Take the interval [0, 1/n] for n in |N. It changes and
>> contains many real numbers for every n.
>
> If you change what n refers to, you are equivocating.
> Equivocating is a fallacy.
>
>
> If you know that n is in N, then you also know many other
> claims about n. Which individual n refers to is not one of them,
> but many other claims.
>
> If, later, you learn that (n-1)*(n-2) = 0, you still don't
> know which individual n refers to. But you will know that
> n is not outside of {2, 3}.

Ouch.
Can we just pretend I wrote "{1, 2}"?

> Take the interval [0, 1/n] for n in |N.  It doesn't change,
> but we know, even without knowing which element of N it is
> that n refers to, that [0, 1/n] contains many real numbers.
>
> We can know claims and reason about claims without knowing
> which specific individual we are referring to, if we only
> make claims that are true of any individual we might be
> referring to.
>
>
> But "changing" what we refer to is chaos. Will we have
> n not equal to n?
>

On 9/1/2021 11:01 AM, Jim Burns wrote:
> On 9/1/2021 11:58 AM, Jim Burns wrote:
>> On 9/1/2021 8:28 AM, WM wrote:
>>> William schrieb
>>> am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>>
>>>> A "gap" that changes is not capable of containing a real number
>>>
>>> Take the interval [0, 1/n] for n in |N. It changes and
>>> contains many real numbers for every n.
>>
>> If you change what n refers to, you are equivocating.
>> Equivocating is a fallacy.
>>
>>
>> If you know that n is in N, then you also know many other
>> claims about n. Which individual n refers to is not one of them,
>> but many other claims.
>>
>> If, later, you learn that (n-1)*(n-2) = 0, you still don't
>> know which individual n refers to. But you will know that
>> n is not outside of {2, 3}.
>
> Ouch.
> Can we just pretend I wrote "{1, 2}"?

how about "Potentially" {1, 2} ?

>
>> Take the interval [0, 1/n] for n in |N.  It doesn't change,
>> but we know, even without knowing which element of N it is
>> that n refers to, that [0, 1/n] contains many real numbers.
>>
>> We can know claims and reason about claims without knowing
>> which specific individual we are referring to, if we only
>> make claims that are true of any individual we might be
>> referring to.
>>
>>
>> But "changing" what we refer to is chaos. Will we have
>> n not equal to n?
>>
>

On 9/1/2021 8:36 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 1. September 2021 um 01:05:46 UTC+2:
>> On 8/31/2021 4:53 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Dienstag, 31. August 2021 um 21:09:42 UTC+2:
>>>> On 8/31/2021 7:53 AM, WM wrote:

>>>>> Sorry, your nice argument is geometric but it is based on
>>>>> similarity. Similarity is not identity.
>>>
>>>> Geometrically, this can be understood as a claim that
>>>> _similar_ triangles Q0C and C0R exist with the point Q
>>>> a distance q from 0 and the point R a distance r from 0,
>>>> and r = 1/q.
>>>
>>> For every definable point Q it is correct,
>>> like for every definable n there is a FISON.
>>
>> We decide to reason about points in a line.
>> We describe a point: some point 1/k is between it and 0.
>> We reason from that description.
>>
>> We decide to reason about the numbers we count with.
>> We describe a number: it has a FISON.
>
> But every finite set of endsegments you can describe has an
> infinite intersection. The infinite set has an empty
> intersection, but you cannot describe most of its endsegments.
> Same with FISONs and points in aline.

I can _make a true claim_ about each end segment.
Same with FISONs and points in a line.

It seems that this is not enough for you.
You want some interaction with each of them.

Nonetheless, the true claim is true, no matter which
end segment, FISON, or point in a line is referred to.

Initially, a claim of this sort is intensely unimpressive.
"An end segment is an end segment." Whco knew?

But, if we preserve the truth of these (unimpressive) claims
in further claims, we can get to very impressive, even
counter-intuitive, truths -- and we will know the further claims
as surely as we know that an end segment is an end segment.

>> What you're calling a contradiction is only you describing
>> _something else_
>
> I describe collectively the number of points in a unit
> interval. I find that all unit intervals have same number
> of points. You find that all intervals have same number of
> points. Therefore, what you call same is not same.

What I call same is there is at least one way to match
the elements, one-to-one. I don't know how I could have been
more clear about that.

Another meaning of "same" is to match one unmatched to the
next natural number until there are no more unmatched.
In this instance, that cannot be what you are I or anyone
means by "same". There is no last point in any interval.

>>> Arithmetic and geometry, the pilars of mathematics,
>>> prove same number of points in every unit interval
>>> and a difference to every other interval.
>
> Argument based upon definabiliyt of all elements of
> infinite sets deletd.

Is arithmetic consistent?
Is division arithmetic?

On 8/31/2021 3:53 PM, WM wrote:
> Jim Burns schrieb am Dienstag, 31. August 2021 um 21:09:42 UTC+2:
>> On 8/31/2021 7:53 AM, WM wrote:
>
>>> Sorry, your nice argument is geometric but it is based on
>>> similarity. Similarity is not identity.
>
>> Geometrically, this can be understood as a claim that
>> _similar_ triangles Q0C and C0R exist with the point Q
>> a distance q from 0 and the point R a distance r from 0,
>> and r = 1/q.
>
> For every definable point Q it is correct, like for every definable n there is a FISON.
>
>> Arithmetic shows an equal number of points, equal number of
>> rationals, equal number of fractions in intervals (0,1] and
>> [1, +inf).
>>
> No, that is too obviously nonsense, to be arithmetic. It is matheology.
>
> Arithmetic and geometry, the pilars of mathematics, prove same number of points in every unit interval and a difference to every other interval.

Fail. In any unit interval there are an infinity of points. did you count ALL of them ?

Provide a proof, not a spoof.

558 Infinite Stupidity Ants
559 Infinite Distance Ants
560 infinite intersection Ants at the crossroads
561 Infinitely Many Ants
562 infinitely many sequences with limit 0 Ants
563 Infinitely Intervals of infinitely Ants
564 Infinitely Missing Ants
565 Infinitely Many Missing Ants
566 Infinitely many Unit Fractional Ants
567 Infinitely Employed Set Emptying Ants
568 Infinitely finite Ants

WM pretended :
> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>> On Tuesday, August 31, 2021 at 5:44:38 PM UTC-3, WM wrote:
>>> William schrieb am Dienstag, 31. August 2021 um 16:51:07 UTC+2:
>>>> On Tuesday, August 31, 2021 at 10:13:33 AM UTC-3, WM wrote:
>>>>> William schrieb am Dienstag, 31. August 2021 um 15:08:58 UTC+2:
>>>>>> On Tuesday, August 31, 2021 at 9:17:24 AM UTC-3, WM wrote:
>>>>>>> William schrieb am Montag, 30. August 2021 um 20:03:10 UTC+2:
>>>>>>> A "gap" is something that can contain an unchanging real number (not at
>>>>>>> "dark real number" whatever that is).
>>>>>>>> A gap is an interval [0,x].
>>>>>>> Yes
>>>>>> Indeed, and x is an unchanging real number (not a "dark real number"
>>>>>> whatever that is).
>>>>> x is either the numerical value, then it is unchanging, but the end of
>>>>> the interval is changing, or x is the end oft the interval, then it is
>>>>> changing (like the largest known prime number).
>>>> Piffle.
>>>> A "gap" does not change.
>>> That would be deplorable for everybody who has gaps in his knowledge. Try
>>> it. It can change.
>>>
>> A "gap" that changes is not capable of containing a real number
>
> Take the interval [0, 1/n] for n in |N. It changes and contains many real
> numbers for every n.

It is a sequence of intervals, one for every natural number.

FromTheRafters schrieb am Mittwoch, 1. September 2021 um 20:56:35 UTC+2:
> WM pretended :
> > William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
> >> On Tuesday, August 31, 2021 at 5:44:38 PM UTC-3, WM wrote:
> >>> William schrieb am Dienstag, 31. August 2021 um 16:51:07 UTC+2:
> >>>> On Tuesday, August 31, 2021 at 10:13:33 AM UTC-3, WM wrote:
> >>>>> William schrieb am Dienstag, 31. August 2021 um 15:08:58 UTC+2:
> >>>>>> On Tuesday, August 31, 2021 at 9:17:24 AM UTC-3, WM wrote:
> >>>>>>> William schrieb am Montag, 30. August 2021 um 20:03:10 UTC+2:
> >>>>>>> A "gap" is something that can contain an unchanging real number (not at
> >>>>>>> "dark real number" whatever that is).
> >>>>>>>> A gap is an interval [0,x].
> >>>>>>> Yes
> >>>>>> Indeed, and x is an unchanging real number (not a "dark real number"
> >>>>>> whatever that is).
> >>>>> x is either the numerical value, then it is unchanging, but the end of
> >>>>> the interval is changing, or x is the end oft the interval, then it is
> >>>>> changing (like the largest known prime number).
> >>>> Piffle.
> >>>> A "gap" does not change.
> >>> That would be deplorable for everybody who has gaps in his knowledge. Try
> >>> it. It can change.
> >>>
> >> A "gap" that changes is not capable of containing a real number
> >
> > Take the interval [0, 1/n] for n in |N. It changes and contains many real
> > numbers for every n.
> It is a sequence of intervals, one for every natural number.

This sequence can be interpreted as the description of a timewise changing interval. Each of these intervals contains real numbers since no definable 1/n is 0.

Regards, WM

Greg Cunt schrieb am Mittwoch, 1. September 2021 um 17:09:59 UTC+2:
> On Wednesday, September 1, 2021 at 2:36:32 PM UTC+2, WM wrote:
>
> > every finite set of endsegments [...] has an infinite intersection. The infinite set [of endsegments] has an empty intersection, but you cannot describe most of its endsegments.
>
> Why do we have to "Mückenheim-describe" them?

In order to see that in many casese it is impossible to describe them. This shows you that they are indescribable or dark.
>
> In math it suffices to DEFINE the notion of an endsegment (and the set of all endsegments) to be able to PROVE certain things concernig endsegements (and sets of endsegments).
>
> For example: We are able to PROVE: Each and every finite set of endsegments has an infinite intersection. Each and every infinite set of endsegments has an empty intersection. Period.

Not period, but question mark: Why is this so? Every definable endsegment has an infinite intersection with ist predecessors. If all infinitely many endsegments were definable, then they all had an infinite intersection. But they are not.

Regards, WM

William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
> On Wednesday, September 1, 2021 at 9:28:50 AM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>
> > > A "gap" that changes is not capable of containing a real number
> > Take the interval [0, 1/n] for n in |N. It changes
> Piffle. If it changes it is not an interval.

Sometimes mathematics is applied to describe closing doors. (That is much more sensible than to describe inaccessible cardinal numbers.)

> And what it contains is not a real number as a real number does not change.

Every definable interval [0, 1/n] contains real numbers.

Regards, WM

Greg Cunt schrieb am Mittwoch, 1. September 2021 um 16:59:48 UTC+2:

> Hint: If we are talking about "changing so-and-so" in math its only a "figure of speech". Mathematical objects don't (actually) "change".

Never heard of fluxions?

Regards, WM

Jim Burns schrieb am Mittwoch, 1. September 2021 um 18:40:47 UTC+2:
> On 9/1/2021 8:36 AM, WM wrote:

> > But every finite set of endsegments you can describe has an
> > infinite intersection. The infinite set has an empty
> > intersection, but you cannot describe most of its endsegments.
> > Same with FISONs and points in aline.
> I can _make a true claim_ about each end segment.
> Same with FISONs and points in a line.
> It seems that this is not enough for you.
> You want some interaction with each of them.

You cannot make a true claim about each endsegment. You only clnclude it from some about which you can make true claims.

> > I describe collectively the number of points in a unit
> > interval. I find that all unit intervals have same number
> > of points. You find that all intervals have same number of
> > points. Therefore, what you call same is not same.
> What I call same is there is at least one way to match
> the elements, one-to-one. I don't know how I could have been
> more clear about that.

There is no way. It would raise a contradiction.
>
> Another meaning of "same" is to match one unmatched to the
> next natural number until there are no more unmatched.
> In this instance, that cannot be what you are I or anyone
> means by "same". There is no last point in any interval.

There is a last positive point before zero. But it is dark.

Regards, WM

On Wednesday, September 1, 2021 at 4:40:34 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:

> > And what [ every element of the collection of all intervals [0,1/n] ] contains is not a real number as a real number does not change.
> Every definable interval [0, 1/n] contains real numbers.
Also every interval [0,1/n] where you cannot write down n contains real numbers. So no interval [0,1/n] is a gap (note there is no element m, of |N_F where [0,1/m] is something that changes) A gap that gets smaller when you want in to is nonsense.

--
William Hughes

On Wednesday, September 1, 2021 at 4:48:02 PM UTC-3, WM wrote:

> There is a last positive point before zero. But it is dark.
>

So whatever it is it is not a number of the form 1/n which n an element of |N_F, nor is it a real number ("dark real numbers" whatever they are are not real numbers)

--
William Hughes

WM has brought this to us :
> William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
>> On Wednesday, September 1, 2021 at 9:28:50 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>>>> A "gap" that changes is not capable of containing a real number
>>> Take the interval [0, 1/n] for n in |N. It changes
>> Piffle. If it changes it is not an interval.
>
> Sometimes mathematics is applied to describe closing doors. (That is much
> more sensible than to describe inaccessible cardinal numbers.)
>
>> And what it contains is not a real number as a real number does not change.
>
> Every definable interval [0, 1/n] contains real numbers.

Rational and irrational, and not just a few either.

Guess how many irrationals? How many rationals?

On Wednesday, September 1, 2021 at 9:33:39 PM UTC+2, WM wrote:

> This sequence can be interpreted as the description of a timewise changing interval.

No. Since there is no "timewise changing interval", no sequence can be "interpreted" as "the description of a timewise changing interval", at least not in math.

"'Contrariwise,' continued Tweedledee, 'if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.'"

(Lewis Carroll, Through the Looking Glass)

On 9/1/2021 3:47 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 1. September 2021 um 18:40:47 UTC+2:
>> On 9/1/2021 8:36 AM, WM wrote:

>>> But every finite set of endsegments you can describe has an
>>> infinite intersection. The infinite set has an empty
>>> intersection, but you cannot describe most of its endsegments.
>>> Same with FISONs and points in aline.
>>
>> I can _make a true claim_ about each end segment.
>> Same with FISONs and points in a line.
>> It seems that this is not enough for you.
>> You want some interaction with each of them.
>
> You cannot make a true claim about each endsegment.

An end segment is an end segment.
And I can say what an end segment is.

An end segment is a non-empty set which is closed upwards.

The set B is closed upwards iff
for all x,y, x < y,
if x is in B, then y is in B.

k is a FISONed natural iff
steppable {0,...,k} exists in which,
for all adjacent i,j, j = i+1.

An end segment of FISONed naturals is a non-empty set
of FISONed naturals which is closed upwards.

For each FISONed natural, there is an end segment E(m+1)
of FISONed naturals which m is NOT in.

Let _Ends_ be an arbitrary collection of end segments of
_FISONed naturals_ and INT(Ends) is its intersection.

If E' is an end segment in the collection Ends,
then INT(Ends) is a subset of E'.

Equivalently,
if INT(Ends) is NOT a subset of E',
then E' is NOT an end segment in the collection Ends.

Assume that INT(Ends) is NOT empty.
Then there is a FISONed natural m in INT(Ends)
Then there is an end segment E(m+1) which m is NOT in.
Then there is an end segment E(m+1) which
INT(ENDS) is NOT a subset of.
Then there is an end segment E(m+1) which
is NOT in the collection Ends.
Then Ends is NOT the collection of all endsegments.

In short,
If INT(Ends) is NOT empty,
then Ends is NOT the collection of all end segments.

Equivalently,
if Ends is the collection of all end segments,
then INT(Ends) is empty.

> You cannot make a true claim about each endsegment.

Clearly I can.

> You only clnclude it from some about which
> you can make true claims.

Concerning end segments, I claim that an end segment
is non-empty and closed upwards.
When I derive further conclusions from that claim,
they will only be conclusions about a set which
is non-empty and closed upwards.
That's enough. That's plenty.
Actually, more would be too much.

I _do not want_ conclusions about a set which is non-empty
and closed upwards to be applied to other sets.
To anyone doing that: _please stop it_

On Wednesday, September 1, 2021 at 3:17:43 PM UTC-7, Greg Cunt wrote:
> On Wednesday, September 1, 2021 at 9:33:39 PM UTC+2, WM wrote:
>
> > This sequence can be interpreted as the description of a timewise changing interval.
> No. Since there is no "timewise changing interval", no sequence can be "interpreted" as "the description of a timewise changing interval", at least not in math.
>
> "'Contrariwise,' continued Tweedledee, 'if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.'"
>
> (Lewis Carroll, Through the Looking Glass)

Dodgson has his own theory of infinitesimals it's one of those things.

It's not so unusually pretty conservative.

But, it's pretty interesting and kind of a mathematical development.

Compared to the quote here which is an idiot's of sorts.

It is a fair rejection of "material implication".

If you might discount the slur inherent in the posts,
or das Hunde ist ein imbecile as it were, otherwise it
accumulates not only a general distaste and indifference
actually even an active dislike and a hate, or sorts.

What you not having the _time_ leaves me noting you late.

On 9/1/2021 4:21 PM, FromTheRafters wrote:
> WM has brought this to us :
>> William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
>>> On Wednesday, September 1, 2021 at 9:28:50 AM UTC-3, WM wrote:
>>>> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2: 
>>>>> A "gap" that changes is not capable of containing a real number
>>>> Take the interval [0, 1/n] for n in |N. It changes
>>> Piffle. If it changes it is not an interval.
>>
>> Sometimes mathematics is applied to describe closing doors. (That is much more sensible than to describe inaccessible cardinal numbers.)
>>
>>> And what it contains is not a real number as a real number does not change.
>>
>> Every definable interval  [0, 1/n] contains real numbers.
>
> Rational and irrational, and not just a few either.
>
> Guess how many irrationals? How many rationals?

Quiz! use the interval [0, 1/n] , where n is a natural number

# irrationals are in the interval ?

1. 17
2. 3,928,665,564
3. infinite #
4. a lot
5. All of the above

How many rationals are in the interval ?

1. infinite #
2. 3,928,665,564
3. a lot
4. 17
5. none of the above

Extra Credit Questions!;

1. Explain if there is a dependence on n, or not.
2. If you cut the interval in half, and only consider 1/2, does that change the above answer ?