def construct(self):
self.force_skipping()
conversation = Conversation(self)
conversation.add_bubble("A co sjednotit všechny ordinály?")
self.revert_to_original_skipping_status()
forti_pic = ImageMobject("BuraliForti1.jpg", use_cache = False)
forti_pic.scale(0.43)
forti_pic.to_corner(UP+LEFT)
ordinal_class = OrdinalClass(color = BLUE).shift(UP)
ordinal_class.next_to(forti_pic, coor_mask = X_MASK)
num = 7
subordinals = VGroup([ordinal_class.copy() for _ in range(num)])
for subord, x in zip(subordinals, np.linspace(0.3, 1, num)):
i = int(x*len(subord.line))
subord.line.remove(*subord.line[i:])
lines_dest = VGroup([subord.line for subord in subordinals]).copy()
bars_dest = VGroup([subord.bars for subord in subordinals]).copy()
subordinals.arrange_submobjects(UP, False, coor_mask = Y_MASK)
subordinals.next_to(conversation.dialog, UP, coor_mask = Y_MASK)
lines = VGroup([subord.line for subord in subordinals])
bars = VGroup([subord.bars for subord in subordinals])
self.wait_to(3)
self.play(FadeIn(subordinals))
self.wait_to(4.5)
self.play(
Transform(lines, lines_dest),
Transform(bars, bars_dest),
)
self.remove(subordinals)
self.add(ordinal_class)
self.wait_to(6)
conversation.add_bubble("Cesare Burali-Forti paradox")
next_bar = ordinal_class.bars[0][0].copy().next_to(ordinal_class)
next_bar.highlight(YELLOW)
self.wait_to(9)
self.play(ShowCreation(next_bar))
brace = BraceText(VGroup(ordinal_class, next_bar), "Nový ordinál", UP)
self.wait_to(15)
self.play(brace.creation_anim())
picture = VGroup(ordinal_class, brace, next_bar)
picture.save_state()
picture.next_to(forti_pic, coor_mask = X_MASK)
self.wait_to(29)
self.play(FadeIn(forti_pic))
self.wait_to(49)
self.play(*map(FadeOut, [
picture,
conversation.dialog,
forti_pic,
]))
def construct(self):
negatives = VGroup(*[self.number_tex(-n) for n in range(9)])
positives = VGroup(*[self.number_tex(n) for n in range(9)])
negatives.shift(UP)
neg_brace = BraceDesc(negatives, "\\aleph_0", UP)
pos_brace = BraceDesc(positives, "\\aleph_0", DOWN)
self.play(ShowCreation(negatives))
self.play(neg_brace.creation_anim())
self.dither()
self.play(ShowCreation(positives))
self.play(pos_brace.creation_anim())
self.dither()
self.play(
positives.shift,
UP,
pos_brace.shift,
UP,
)
self.remove(positives[0])
self.play(
FadeOut(pos_brace),
neg_brace.shift_brace,
VGroup(negatives, negatives[-1].copy().move_to(positives[-1])),
)
self.dither()
return
conversation = Conversation(self)
conversation.add_bubble("Hey, what happened to the merged zero?")
self.dither()
conversation.add_bubble("The size is still countable infinite.")
self.dither()
def construct(self):
self.h_space = 2
self.colors = color_gradient(
[YELLOW, GREEN, BLUE, DARK_BLUE, DARK_GRAY], 20)
numbers = self.make_numbers()
self.play(
ShowCreation(numbers[0]),
ShowCreation(numbers[1]),
)
brace = BraceDesc(numbers[0], "\\aleph_0", DOWN)
brace.next_to(numbers[1], DOWN, coor_mask=Y_MASK, buff=0.3)
self.play(brace.creation_anim())
skip = False
denominator = 2
parts = []
while True:
parts.append(numbers[0].copy())
self.add(parts[-1])
self.play(VGroup(numbers, brace).shift, 0.9 * DOWN)
self.remove(numbers)
if denominator == 6: break
numbers = self.make_numbers(denominator, numbers)
ori_numbers = self.make_numbers(denominator - 1, numbers)
if skip:
offset = (len(ori_numbers[1]) // 2 + 1) % 2
for label in ori_numbers[1][offset::2]:
label.set_fill(opacity=0)
if denominator == 4: skip = True
if skip:
offset = (len(numbers[1]) // 2 + 1) % 2
for label in numbers[1][offset::2]:
label.set_fill(opacity=0)
self.play(ReplacementTransform(ori_numbers, numbers))
denominator += 1
self.dither()
self.remove(brace, numbers)
while denominator < len(self.colors) + 1:
part = self.make_dots(denominator)
part.move_to(parts[-1])
part.shift(0.5 * DOWN)
parts.append(part)
denominator += 1
parts.reverse()
parts = VGroup(*parts)
parts_dest = parts.copy()
for part in parts_dest:
part.center()
self.play(Transform(parts, parts_dest), run_time=2)
self.dither()
rationals = TexMobject('|', "\\text{Rational Numbers}",
"| = \\aleph_0")
rationals.shift(0.5 * DOWN - rationals[1].get_edge_center(UP))
self.play(Write(rationals[1]))
self.dither()
dots7 = self.make_dots(7)
dots7.shift(2 * DOWN)
center = len(dots7) // 2
pi_dot = dots7[center + 22]
pi_label = TexMobject("\\frac{22}{7}")
pi_label.next_to(pi_dot, DOWN)
self.play(Write(pi_label))
self.dither()
self.play(ShowCreation(dots7))
self.play(FocusOn2(pi_dot, scale=3))
dots7.remove(pi_dot)
self.play(FadeOut(dots7))
self.play(VGroup(pi_dot, pi_label).shift, 2 * UP)
self.dither()
self.play(FadeOut(VGroup(pi_dot, pi_label)))
self.play(Write(VGroup(rationals[0], rationals[2])))
conversation1 = Conversation(self)
conversation1.add_bubble("Are all sets countable?")
self.dither()
conversation2 = Conversation(self, start_answer=True)
conversation2.add_bubble("Of course not.")
self.dither()
self.prepare_overall_picture()
self.overall_picture.to_edge(UP, buff=0.4)
self.overall_picture.remove(self.uncountable_brace)
self.play(FadeIn(self.overall_picture))
self.dither()
self.play(self.uncountable_brace.creation_anim())
self.dither()
real_size = TexMobject("|\\mathbb R|")
real_size.set_color(GREEN)
real_size.next_to(self.sizes, buff=1)
self.play(Write(real_size))
self.dither()
def construct(self):
# self.force_skipping()
# left-closed, right-open, two intervals
interval1 = SemiOpen()
interval2 = interval1.copy()
VGroup(interval1, interval2).arrange_submobjects(buff=1)
self.play(
FadeIn(interval1),
FadeIn(interval2),
)
self.wait_to(11)
self.play(
interval1.move_to,
LEFT,
interval2.move_to,
RIGHT,
)
self.remove(interval1.second_point)
self.play(interval2.first_point.scale_in_place, 0)
interval = SemiOpen(2 * LEFT, 2 * RIGHT)
self.remove(interval1, interval2)
self.add(interval)
self.dither()
interval_dest = SemiOpen(LEFT, RIGHT)
self.play(Transform(interval, interval_dest))
self.wait_to(22)
self.play(FadeOut(interval))
interval1 = SemiOpen()
interval2 = interval1.copy()
VGroup(interval1, interval2).arrange_submobjects(buff=1)
for interval in [interval1, interval2]:
interval.stretch_in_place(-1, 0)
# left-open, right-closed, two intervals
self.play(
FadeIn(interval1),
FadeIn(interval2),
)
self.play(
interval2.move_to,
RIGHT,
interval1.move_to,
LEFT,
)
self.remove(interval2.second_point)
self.play(interval1.first_point.scale_in_place, 0)
interval = SemiOpen(2 * RIGHT, 2 * LEFT)
self.remove(interval1, interval2)
self.add(interval)
interval_dest = SemiOpen(RIGHT, LEFT)
self.play(Transform(interval, interval_dest))
self.wait_to(27.5)
# left-open, right-closed, omega
omega = OrdinalOmega()
omega.highlight(DARK_GREY)
interval_copies = []
for bar in omega:
interval_copy = SemiOpen(RIGHT * bar.x1,
RIGHT * bar.x0,
dot_radius=0.1 * bar.height)
interval_copies.append(interval_copy)
final_interval = SemiOpen(RIGHT * omega.x0, RIGHT * omega.x1)
interval_copies.reverse()
interval_copies = VGroup(interval_copies)
VGroup(omega, interval_copies, final_interval).shift(UP)
self.play(
FadeIn(omega),
interval.shift,
0.8 * DOWN,
)
interval_copies_src = interval_copies.copy()
for interval_copy in interval_copies_src:
interval_copy.scale_in_place(0)
self.play(ReplacementTransform(interval_copies_src, interval_copies))
to_shrink = []
for interval_copy in interval_copies:
self.remove(interval_copy.second_point)
to_shrink.append(interval_copy.first_point)
self.add(interval_copies[-1].second_point)
to_shrink = VGroup(to_shrink)
to_shrink_ori = to_shrink.copy()
shrinked = to_shrink.copy()
for dot in shrinked:
dot.scale_in_place(0)
self.play(
Transform(to_shrink, shrinked),
UnapplyMethod(final_interval.second_point.scale_in_place, 0),
run_time=0.5,
)
self.remove(interval_copies)
self.add(final_interval.line, interval_copies[-1].second_point)
self.add_foreground_mobjects(final_interval.second_point)
self.wait_to(42)
# left-closed, right-open, omega
self.play(Rotate(interval, in_place=True)) # 43
self.dither() # 44
self.play(
Transform(to_shrink, to_shrink_ori),
run_time=0.5,
) # 44.5
interval_copies_dest = interval_copies.copy()
bg, bg_dest, fg, fg_dest = [], [], [], []
for interval_copy_dest in interval_copies_dest:
interval_copy_dest.scale_in_place(-1)
bg_dest += [
interval_copy_dest.line, interval_copy_dest.second_point
]
fg_dest.append(interval_copy_dest.first_point)
for interval_copy in interval_copies:
bg += [interval_copy.line, interval_copy.second_point]
fg.append(interval_copy.first_point)
bg, bg_dest, fg, fg_dest = map(VGroup, (bg, bg_dest, fg, fg_dest))
self.remove(final_interval)
self.add_foreground_mobjects(final_interval.second_point)
self.play(
Transform(bg, bg_dest, path_arc=np.pi),
Transform(fg, fg_dest, path_arc=np.pi),
)
self.remove(bg)
self.add(final_interval)
shrinked = to_shrink.copy()
for dot in shrinked:
dot.scale_in_place(0)
self.play(
Transform(to_shrink, shrinked),
run_time=0.5,
)
# left-closed, right-open, omega*2
interval_template = interval # some renaming
interval = final_interval
omega_cdot_2 = OrdinalFiniteProd(OrdinalOmega, 2)
omega_cdot_2.set_color(DARK_GREY)
intervals2 = [
SemiOpen(RIGHT * ordinal.x0, RIGHT * ordinal.x1)
for ordinal in omega_cdot_2[0], omega_cdot_2[1][0], omega_cdot_2[1]
[1], omega_cdot_2
]
intervals2.append(
SemiOpen(RIGHT * omega_cdot_2.x0, RIGHT * omega_cdot_2[1][1].x1))
VGroup(omega_cdot_2, intervals2).shift(UP)
self.wait_to(50)
self.play(
ReplacementTransform(omega, omega_cdot_2[0]),
Transform(interval, intervals2[0]),
)
self.revert_to_original_skipping_status()
self.wait_to(52.5)
self.remove_foreground_mobjects(interval.second_point)
interval2_1_src = intervals2[1].copy()
interval2_1_src.set_color(BLACK).shift(RIGHT)
self.play(ShowCreation(omega_cdot_2[1][0]), Animation(interval),
ReplacementTransform(interval2_1_src, intervals2[1]))
self.remove(interval.second_point)
interval.line.scale_about_point(1.1,
interval.line.get_edge_center(LEFT))
self.play(intervals2[1].first_point.scale_in_place, 0, run_time=0.5)
self.wait_to(54.5)
interval2_2_src = intervals2[2].copy()
interval2_2_src.set_color(BLACK).shift(RIGHT)
self.play(ShowCreation(omega_cdot_2[1][1]), Animation(intervals2[1]),
ReplacementTransform(interval2_2_src, intervals2[2]))
self.remove(interval, intervals2[1], intervals2[2])
interval = intervals2[4]
self.add(interval)
self.play(intervals2[2].first_point.scale_in_place, 0, run_time=0.5)
self.remove(intervals2[2].first_point)
self.wait_to(56)
self.play(
FadeIn(VGroup(omega_cdot_2[1][2:]),
submobject_mode="lagged_start"),
Transform(interval, intervals2[3]),
)
self.wait_to(60 + 7)
conversation = Conversation(self)
conversation.add_bubble(
"Neměl by být výsledek delší než první činitel?")
self.wait_to(60 + 20.5)
conversation.add_bubble("Intervaly nejsou dobře uspořádané.")
self.wait_to(60 + 49)
self.play(FadeOut(VGroup(omega_cdot_2, interval, conversation.dialog)))
def construct(self, fast_play=False):
#self.force_skipping()
omega = OrdinalOmega()
omega_plus_5 = OrdinalSum(OrdinalOmega, 0.75,
lambda **kwargs: OrdinalFinite(5, **kwargs))
omega_times_3 = OrdinalFiniteProd(OrdinalOmega, 3)
omega_pow_3 = make_ordinal_power(3, q=(0.7, 0.84, 0.84))
ordinals = VGroup(omega, omega_plus_5, omega_times_3, omega_pow_3)
ordinals.stretch(0.4, 1)
ordinals.arrange_submobjects(DOWN, aligned_edge=LEFT, center=False)
ordinals.shift(-Y_MASK * ordinals.get_center())
dots = TexMobject("\\vdots")
dots.next_to(ordinals, DOWN)
if fast_play:
self.add(ordinals, dots)
else:
self.play(ShowCreation(omega))
self.wait_to(6)
self.play(ShowCreation(omega_plus_5))
self.wait_to(7.3)
self.play(ShowCreation(omega_times_3))
self.wait_to(8.7)
self.play(ShowCreation(omega_pow_3))
self.play(FadeIn(dots))
result = OrdinalFiniteProd(OrdinalOmega, 5, x1=8)
result_stretched = result.copy()
result_stretched.stretch(
omega[0].get_height() / result_stretched[0][0].get_height(), 1)
al_omega = result_stretched[0].copy()
al_omega.shift(omega[0].get_center() - al_omega[0].get_center())
al_omega_plus_5 = VGroup(result_stretched[0].copy())
al_omega_plus_5.shift(omega_plus_5[0][0].get_center() -
al_omega_plus_5[0][0].get_center())
al_five = omega_plus_5[1].copy()
for bar, dest in zip(al_five, result[1]):
bar.move_to(dest, coor_mask=X_MASK)
al_omega_plus_5.add(al_five)
al_omega_times_3 = VGroup(*result_stretched[:3]).copy()
al_omega_times_3.shift(omega_times_3[0][0].get_center() -
al_omega_times_3[0][0].get_center())
result_in_pow_3 = result_stretched.copy()
result_in_pow_3.shift(omega_pow_3[0][0][0].get_center() -
result_in_pow_3[0][0].get_center())
al_omega_pow_3 = omega_pow_3.copy()
al_omega_pow_3.next_to(result_in_pow_3, buff=0)
al_omega_pow_3[0].add_to_back(*result_in_pow_3.submobjects)
al_ordinals = VGroup(al_omega, al_omega_plus_5, al_omega_times_3,
al_omega_pow_3)
if not fast_play: self.wait_to(17.5)
self.play(ReplacementTransform(ordinals, al_ordinals))
al_ordinals.remove(al_omega_pow_3)
al_ordinals.add(result_in_pow_3)
result_bottom = result.copy()
result_bottom.stretch(0, 1)
result_bottom = VGroup(*[result_bottom.copy() for _ in range(5)])
result_bottom.arrange_submobjects(DOWN, buff=0.3, center=False)
result_bottom.to_edge(DOWN, buff=0)
al_ordinals.add(result_bottom)
mrg_omega = result[0].copy()
mrg_omega_plus_5 = VGroup(result[0], VGroup(*result[1][:5])).copy()
mrg_omega_times_3 = VGroup(*result[:3]).copy()
mrg_ordinals = VGroup(
mrg_omega,
mrg_omega_plus_5,
mrg_omega_times_3,
result.copy(),
VGroup(result.copy()),
)
if not fast_play: self.wait_to(19)
self.play(
FadeOut(dots),
ReplacementTransform(al_ordinals, mrg_ordinals),
)
self.remove(mrg_ordinals)
self.add(result)
if fast_play: return result
brace = BraceDesc(result, "\\omega_1", UP)
self.wait_to(20.2)
self.play(brace.creation_anim())
self.wait_to(22)
conversation = Conversation(self)
conversation.add_bubble(
"Ale podle kapitoly 5 si nemůžeme vzít všechny ordinály.")
self.wait_to(29.5)
conversation.add_bubble("To ne, ale všechny spočetné můžeme.")
self.wait_to(39)
self.add_foreground_mobjects(conversation.dialog)
self.play(FadeOut(VGroup(brace, result)))
omega = OrdinalOmega()
omega.shift(2 * UP)
self.play(ShowCreation(omega))
self.wait_to(48.5)
self.play(FadeOut(conversation.dialog))
def construct(self):
conversation = Conversation(self, start_answer=True)
conversation.add_bubble("Existuje množina všech množin?")
conversation.add_bubble("Jasně, proč by neměla?")
self.dither()
conversation.add_bubble(
"A co množina všech množin, které neobsahují samy sebe?")
self.dither(2)
conversation.add_bubble("Tak z té první některé vyhodíme.")
self.dither()
conversation.add_bubble("Obsahuje taková množina samu sebe?")
conversation.add_bubble("Jejda...")
self.wait_to(12.5)
def construct(self):
num = 5
ordinals_cnt_base = OrdinalFiniteProd(OrdinalOmega,
num,
x1=-4 + num * 2.5)
ordinals_cnt = VGroup(
[VGroup(ordinals_cnt_base[:i]).copy() for i in range(1, num + 1)])
ordinals_cnt.save_state()
ordinals_cnt.arrange_submobjects(DOWN, center=False, coor_mask=Y_MASK)
ordinals_cnt.to_edge(UP)
self.play(ShowCreation(ordinals_cnt))
#self.wait_to(7)
self.dither()
self.play(ordinals_cnt.restore)
self.remove(ordinals_cnt)
self.add(ordinals_cnt_base)
omega1 = Omega1()
brace = BraceDesc(omega1, "\\omega_1", DOWN)
omega1.line.save_state()
omega1.line.behind_edge(RIGHT)
omega1_bars_src = make_ordinal_power(2)
omega1_bars_src.next_to(ordinals_cnt_base)
omega1_bars_src.add_to_back(*ordinals_cnt_base)
brace.desc.highlight(BLUE)
brace.save_state()
brace.shift_brace(omega1_bars_src)
brace.highlight(BLACK)
self.play(
brace.restore,
omega1.line.restore,
ReplacementTransform(omega1_bars_src, omega1.bars),
)
#self.wait_to(11)
self.dither()
aleph1 = TexMobject("\\aleph_1", "=|", "\omega_1", "|")
aleph1[2].highlight(BLUE)
aleph1[0].highlight(RED)
aleph1.shift(brace.desc.get_center() - aleph1[2].get_center())
aleph1.remove(aleph1[2])
self.play(Write(aleph1))
#self.wait_to(15)
self.dither()
ordinals_a1 = VGroup([omega1.copy() for _ in range(num)])
aleph1[0].save_state()
aleph1[0].next_to(omega1, LEFT, coor_mask=X_MASK)
self.play(
omega1.to_edge,
UP,
FadeOut(VGroup(brace, aleph1[1:])),
MoveFromSaved(aleph1[0]),
)
#self.wait_to(18.5)
self.dither()
aleph1 = aleph1[0]
scale_center = np.array(ordinals_a1[0].line.get_start())
for i, o in enumerate(ordinals_a1):
o.line.scale(1 + i * 0.2, about_point=scale_center)
ordinals_a1_lines_dest = VGroup([o.line for o in ordinals_a1]).copy()
ordinals_a1_bars_dest = VGroup([o.bars for o in ordinals_a1]).copy()
ordinals_a1.arrange_submobjects(DOWN, False, coor_mask=Y_MASK)
ordinals_a1.to_edge(UP)
ordinals_a1_lines = VGroup([o.line for o in ordinals_a1])
ordinals_a1_bars = VGroup([o.bars for o in ordinals_a1])
self.remove(omega1)
self.add(ordinals_a1[0])
self.play(FadeIn(VGroup(ordinals_a1[1:])))
self.play(
Transform(ordinals_a1_lines, ordinals_a1_lines_dest),
Transform(ordinals_a1_bars, ordinals_a1_bars_dest),
)
self.remove(*ordinals_a1[:-1])
omega2_src = ordinals_a1[-1]
omega2 = LongOrdinal(ordinal_end=0.3, line_start=0.2)
brace = BraceDesc(omega2, "\\omega_2", DOWN)
brace.desc.highlight(BLUE)
brace.save_state()
brace.shift_brace(omega2_src)
brace.highlight(BLACK)
self.play(
brace.restore,
ReplacementTransform(omega2_src, omega2),
)
#self.wait_to(24.5)
self.dither()
aleph2 = TexMobject("\\aleph_2", "=|", "\omega_2", "|")
aleph2[2].highlight(BLUE)
aleph2[0].highlight(RED)
aleph2.shift(brace.desc.get_center() - aleph2[2].get_center())
aleph2.remove(aleph2[2])
self.play(Write(aleph2))
#self.wait_to(28)
self.dither()
ordinal_fin = OrdinalFinite(7, x1=SPACE_WIDTH + 1).highlight(BLUE)
ord_labels = VGroup([
TexMobject("\\omega_{}".format(i + 1)).highlight(BLUE).next_to(
bar, UP) for i, bar in enumerate(ordinal_fin[:4])
])
card_labels = VGroup([
TexMobject("\\aleph_{}".format(i + 1)).highlight(RED).next_to(
bar, DOWN) for i, bar in enumerate(ordinal_fin[:4])
])
self.play(
FadeOut(VGroup(aleph2[1:], brace, omega2)),
ReplacementTransform(aleph1, card_labels[0]),
ReplacementTransform(aleph2[0], card_labels[1]),
)
for i, t in enumerate((29, 31, 35.5, 38)):
#self.wait_to(t)
self.play(FadeIn(ord_labels[i]), ShowCreation(ordinal_fin[i]))
if i >= 2: self.play(FadeIn(card_labels[i]))
#self.wait_to(41)
self.play(ShowCreation(VGroup(ordinal_fin[4:], rate_func=rush_into)))
omega = OrdinalOmega().highlight(BLUE)
omega_src = omega.copy()
omega_src.next_to(ordinal_fin, buff=1)
omega_src.add_to_back(*ordinal_fin)
label1 = VGroup(ord_labels[0], card_labels[0])
labels_to_fade = VGroup(list(zip(ord_labels[1:], card_labels[1:])))
labels_to_fade.save_state()
for l, bar in zip(labels_to_fade, omega[1:4]):
l.move_to(bar, coor_mask=X_MASK)
l.highlight(BLACK)
limit_bar = omega[0].copy().next_to(omega)
limit_lord = TexMobject("\\omega_\\omega").highlight(BLUE).next_to(
limit_bar, UP)
limit_lcard = TexMobject("\\aleph_\\omega").highlight(RED).next_to(
limit_bar, DOWN)
limit_g = VGroup(limit_bar, limit_lord, limit_lcard)
limit_g.next_to(omega, coor_mask=X_MASK)
limit_g.save_state()
limit_g.next_to(omega_src, coor_mask=X_MASK)
self.play(
MoveFromSaved(labels_to_fade),
ReplacementTransform(omega_src, omega),
Animation(label1),
limit_g.restore,
)
self.remove(labels_to_fade)
#self.wait_to(52)
plus1_bar = limit_bar.copy()
plus1_label = TexMobject("\\omega_{\\omega+1}").highlight(
BLUE).next_to(plus1_bar, UP)
plus1_g = VGroup(plus1_bar, plus1_label)
plus1_g.next_to(limit_g, coor_mask=X_MASK, buff=0.5)
self.play(FadeIn(plus1_label), ShowCreation(plus1_bar))
#self.wait_to(55)
omega_pow = make_ordinal_power(2, q=(0.8, 0.9, 0.9)).highlight(BLUE)
omega_pow_src = omega_pow.copy()
omega_pow_src.behind_edge(RIGHT)
omega_pow_src[0].add_to_back(limit_bar, plus1_bar)
omega_pow_src.add_to_back(omega)
limit_g.remove(limit_bar)
limit_g.save_state()
plus1_label.save_state()
limit_g.move_to(omega_pow[1][0], coor_mask=X_MASK)
plus1_label.move_to(omega_pow[1][1], coor_mask=X_MASK).highlight(BLACK)
pow_bar = limit_bar.copy()
pow_label = TexMobject("\\omega_{\\omega\\cdot\\omega}").highlight(
BLUE).next_to(pow_bar, UP)
pow_g = VGroup(pow_bar, pow_label)
pow_g.next_to(omega_pow, coor_mask=X_MASK, buff=-.2)
pow_g.save_state()
pow_g.next_to(omega_pow_src, coor_mask=X_MASK)
self.play(
MoveFromSaved(plus1_label),
ReplacementTransform(omega_pow_src, omega_pow),
MoveFromSaved(limit_g),
pow_g.restore,
)
self.remove(plus1_label)
#self.wait_to(59)
self.dither()
labels_to_fade = VGroup(label1, limit_g, pow_label)
labels_to_fade.save_state()
labels_to_fade.highlight(BLACK)
omega1 = Omega1(color=BLUE)
omega1_bar = pow_bar.copy()
omega1_label = TexMobject("\\omega_{\\omega_1}").highlight(
BLUE).next_to(omega1_bar, UP)
omega1_g = VGroup(omega1_bar, omega1_label)
omega1_g.shift(omega1.line.get_end() - omega1_bar.get_center())
omega1_g.add(omega1.line)
omega1_g.save_state()
omega1_g.behind_edge(RIGHT)
limit_g.move_to(omega1.bars[1][0], coor_mask=X_MASK)
pow_label.move_to(omega1.bars[-1], coor_mask=X_MASK)
pow_bar.save_state()
pow_bar.scale(0)
pow_bar.set_stroke(width=0)
pow_bar.move_to(omega1.bars[-1])
self.play(
MoveFromSaved(labels_to_fade, remover=True),
omega1_g.restore,
MoveFromSaved(pow_bar, remover=True),
ReplacementTransform(omega_pow, omega1.bars),
)
#self.wait_to(60+7.5)
self.dither()
ordinal_fin = OrdinalFinite(6, x1=SPACE_WIDTH + 1).highlight(BLUE)
omega1.save_state()
omega1.stretch_about_point(0, 0, ordinal_fin[0].get_center())
labels = VGroup([
TexMobject((i + 1) * "\\omega_{" + "\\omega_1" +
(i + 1) * "}").next_to(bar, UP)
for i, bar in enumerate(ordinal_fin[:3])
])
labels.highlight(BLUE)
next_bar = VGroup(labels[1], ordinal_fin[1])
next_bar.save_state()
next_bar.shift(8 * RIGHT)
self.play(
MoveFromSaved(omega1, remover=True),
next_bar.restore,
ReplacementTransform(omega1_label, labels[0]),
ReplacementTransform(omega1_bar, ordinal_fin[0]),
)
#self.wait_to(60+11)
self.play(FadeIn(labels[2]), ShowCreation(ordinal_fin[2]))
#self.wait_to(60+13)
self.play(ShowCreation(VGroup(ordinal_fin[3:]), rate_func=rush_into))
#self.wait_to(60+21.5)
self.dither(2)
conversation = Conversation(self)
conversation.add_bubble("What about the union of all the ordinals?")
self.play(FadeOut(VGroup(labels, ordinal_fin)),
Animation(conversation.dialog))
def construct(self):
conversation = Conversation(self, start_answer=True)
conversation.add_bubble("Is there a set of all sets?")
conversation.add_bubble("Sure, why shouldn't be?")
self.dither()
conversation.add_bubble(
"What about the set of all sets not containing themselves?")
self.dither()
conversation.add_bubble(
"Just discard some elements of the previous one?")
self.dither()
conversation.add_bubble("Does such a set contain itself?")
conversation.add_bubble("Oops...")