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):
omega = OrdinalOmega()
omega_brace = BraceDesc(omega, "\\omega", UP)
self.add(omega_brace)
omega, _ = self.play_infinite_supremum(waits=(4, 5.1))
fin_index = 12
bar = omega[fin_index].copy()
bar.highlight(YELLOW)
arrow = Arrow(
bar.get_edge_center(DOWN) + DOWN,
bar.get_edge_center(DOWN),
)
label = TextMobject("konečný ordinál").next_to(arrow, DOWN)
self.wait_to(16)
self.play(
ShowCreation(bar),
ShowCreation(arrow),
FadeIn(label),
)
finite_brace = Brace(VGroup(omega[:fin_index]), DOWN)
brace_label = label.copy()
finite_brace.put_at_tip(brace_label)
self.wait_to(20.3)
self.play(
GrowFromCenter(finite_brace),
ReplacementTransform(label, brace_label),
)
terminal_segment_brace = BraceText(
Line(omega[fin_index + 1].get_center() + DOWN,
omega.get_edge_center(RIGHT)), "nekonečno")
self.wait_to(25.5)
self.play(terminal_segment_brace.creation_anim())
self.wait_to(41)
self.play(
FadeOut(
VGroup(
finite_brace,
brace_label,
terminal_segment_brace,
bar,
arrow,
)))
def construct(self):
series = VideoSeries(num_videos = 16).to_edge(UP)
self.add(series)
brace = BraceText(series[:7], "Theory of Infinity")
self.play(brace.creation_anim())
self.dither()
self.play(brace.change_brace_desc, series[7:14], "Foundations of Mathematics")
self.dither()
brace_dest = brace.copy()
brace_dest.change_brace_desc(series[14:16], "Tools")
brace_dest.desc.to_edge(RIGHT)
self.play(ReplacementTransform(brace, brace_dest))
brace = brace_dest
self.dither()
def construct(self):
#self.force_skipping()
subset = self.al_count_subset
self.play(FadeIn(subset))
self.wait_to(5)
self.play(self.count_brace.creation_anim())
self.dither() # 6
self.play(
Transform(subset, self.shifted_count_subset),
self.count_brace.shift_brace,
self.shifted_count_subset,
) # 7
supremum = Line(ORIGIN, DOWN).next_to(subset, buff=0)
supremum.set_color(YELLOW)
self.wait_to(14)
self.play(ShowCreation(supremum))
self.wait_to(15.5)
identity = TexMobject("\\aleph_0\\cdot\\aleph_0 = \\aleph_0")
identity.next_to(supremum, buff=1)
self.play(FadeIn(identity))
supremum_in_ori = supremum.copy()
supremum_in_ori.move_to(self.omega1, coor_mask=Y_MASK)
self.wait_to(27.2)
self.play(ReplacementTransform(supremum.copy(), supremum_in_ori))
self.wait_to(30)
self.play(FadeOut(identity))
brace_base = Line(
supremum_in_ori.get_edge_center(DOWN),
self.omega1[0].get_edge_center(RIGHT),
)
brace_unused = BraceText(brace_base, "nespočetno")
self.play(brace_unused.creation_anim())
self.wait_to(44.5)
self.play(
FadeOut(
VGroup(brace_unused, subset, supremum, supremum_in_ori,
self.count_brace)))
seq = OrdinalOmega(x0=-1, x1=2)
seq.set_color(GREEN)
seq.stretch(0.5, 1)
seq.shift(self.shift * UP)
self.play(ShowCreation(seq))
supremum.next_to(seq, buff=0)
self.wait_to(53.5)
self.play(ShowCreation(supremum))
self.wait_to(60 + 7)
self.play(FadeOut(VGroup(seq, supremum)))
seq = OrdinalOmega(x0=-1, x1=4)
seq.set_color(GREEN)
seq.stretch(0.5, 1)
seq.shift(self.shift * UP)
self.play(ShowCreation(seq))
self.revert_to_original_skipping_status()
self.wait_to(60 + 17)
self.omega1.add(seq)
stretched_omega1 = self.omega1.copy()
stretched_omega1.stretch_about_point(0.6, 0,
self.omega1.get_edge_center(LEFT))
added_line = Line(stretched_omega1.get_edge_center(RIGHT),
self.omega1.get_edge_center(RIGHT),
color=WHITE)
self.add(added_line)
supremum.move_to(added_line.get_edge_center(LEFT))
brace_unc = BraceText(VGroup(supremum, added_line), "nespočetno")
brace_src = brace_unc.brace.copy()
brace_src.stretch_about_point(0, 0, added_line.get_edge_center(RIGHT))
self.play(
Transform(self.omega1, stretched_omega1),
ReplacementTransform(brace_src, brace_unc.brace),
FadeIn(brace_unc.desc),
)
self.play(ShowCreation(supremum))
self.wait_to(60 + 35)
def construct(self):
title = TextMobject("Hypotéza kontinua").to_edge(UP)
question = TextMobject(
"Je $|\mathbb R|$ nejmenší nespočetná mohutnost?")
question.shift(DOWN)
self.add(title)
cardinalities = TexMobject("0, 1, 2, \ldots", "\\aleph_0", "\\aleph_1",
"=", "|\\mathbb R|", "?")
cardinalities.set_color(GREEN)
cardinalities[1].next_to(cardinalities[0], coor_mask=X_MASK, buff=1)
VGroup(cardinalities[2:]).next_to(cardinalities[1],
coor_mask=X_MASK,
buff=0.5)
cardinalities.next_to(title, DOWN)
count_brace = BraceText(VGroup(cardinalities[:2]), "spočetné")
self.play(
count_brace.creation_anim(),
*map(FadeIn,
[cardinalities[0], cardinalities[1], cardinalities[4]]))
self.play(Write(question))
arrow_end = (cardinalities[1].get_center() +
cardinalities[4].get_center()) / 2
arrow = Arrow(arrow_end + DOWN, arrow_end)
why_not_answer = TextMobject("Proč by tu mělo\\\\ cokoli být?",
alignment="\\raggedright")
why_not_answer.next_to(arrow, DOWN, aligned_edge=LEFT)
why_not_answer.shift(0.5 * LEFT)
self.wait_to(22)
self.play(ShowCreation(arrow), FadeIn(why_not_answer))
self.wait_to(30.5)
self.play(FadeOut(VGroup(arrow, why_not_answer))) # 31.5
why_answer = VGroup(cardinalities[3], cardinalities[5])
why_answer.set_color(WHITE)
self.play(Write(cardinalities[2])) # omega1
self.play(Write(why_answer))
omega1 = Omega1()
make_half_ordinal(omega1[1])
reals = NumberLine(x_min=-4, x_max=4)
reals.set_color(WHITE)
reals.to_edge(DOWN)
omega1.next_to(reals, UP, buff=2)
self.wait_to(50)
self.play(FadeOut(question))
self.play(FadeIn(reals), FadeIn(omega1))
positions_num = 50
positions = np.expand_dims(np.linspace(-4, 4, positions_num),
1) * RIGHT
positions_real = positions + reals.get_edge_center(UP) + UP * 0.1
positions_omega1 = positions + omega1.get_edge_center(
DOWN) + DOWN * 0.1
positions_omega1 = list(positions_omega1)
random.shuffle(positions_omega1)
matching = VGroup([
Line(start, end)
for start, end in zip(positions_omega1, positions_real)
])
matching.set_color(BLUE)
self.play(ShowCreation(
matching,
submobject_mode="all_at_once",
))
self.wait_to(60 + 1)
self.play(FadeOut(matching))
self.wait_to(60 + 10.5)
def construct(self):
#self.force_skipping()
shift = self.shift
# countable subset
self.play(Transform(self.count_subset, self.shifted_count_subset))
self.wait_to(5.1)
self.play(Transform(self.count_subset, self.al_count_subset))
self.wait_to(12)
self.play(self.count_brace.creation_anim())
self.wait_to(17)
self.play(FadeOut(VGroup(self.count_brace, self.count_subset)))
# uncountable subset
unc_initial = LimitSubOrdinal([subord[0] for subord in self.omega1[1]])
unc_initial = unc_initial.copy()
unc_line = DashedLine(self.omega1[0].get_edge_center(LEFT),
self.omega1[0].get_edge_center(RIGHT))
unc_line.gradient_highlight(BLACK, WHITE)
start_x = unc_initial.get_edge_center(RIGHT)[0]
unc_line.submobjects = filter(
lambda l: l.get_edge_center(LEFT)[0] > start_x,
unc_line.submobjects)
unc_subset = VGroup(unc_initial, unc_line)
self.play(VGroup(unc_initial, unc_line).shift, 2 * shift * DOWN)
self.revert_to_original_skipping_status()
self.dither()
al_unc_subset = Omega1().shift(shift * DOWN)
al_unc_line, al_unc_initial = al_unc_subset
ini_index = 9
al_unc_line_parts = []
# split gradient line to larger parts
line_len = (unc_line[2].get_center() - unc_line[1].get_center())[0]
remaining = len(unc_line) - (ini_index + 1) - 1
next_split = al_unc_line.get_edge_center(LEFT)[0]
for grad_part in al_unc_line:
if remaining > 0 and grad_part.get_center()[0] >= next_split:
next_split += line_len
al_unc_line_parts.append([])
al_unc_line_parts[-1].append(grad_part)
al_unc_line_parts = VGroup(al_unc_line_parts)
self.play(
Transform(
VGroup(unc_line[ini_index + 1:]),
al_unc_line_parts,
), Transform(unc_initial, al_unc_initial[0]),
Transform(
VGroup(unc_line[:ini_index]),
VGroup(al_unc_initial[1:ini_index + 1]),
),
Transform(
unc_line[ini_index],
VGroup(unc_initial[ini_index + 1:]),
))
self.remove(*self.mobjects_from_last_animation)
unc_subset = al_unc_subset
self.add(unc_subset)
unc_brace = BraceText(unc_subset, "stejná")
self.play(unc_brace.creation_anim())
cardinality = TexMobject("|", "\\omega_1", "|=\\aleph_1")
cardinality.shift(self.omega1_brace.desc.get_center() -
cardinality[1].get_center())
cardinality.remove(cardinality[1])
self.wait_to(36)
self.play(Write(cardinality))
self.wait_to(46)
class LeastInfinite(Scene):
def setup(self):
self.shift = 1.1
shift = self.shift
self.omega = OrdinalOmega()
self.omega_brace = BraceDesc(self.omega, "\\omega", UP)
self.omega.set_color(BLUE)
self.add(self.omega, self.omega_brace)
# finite subset
self.subset_fin = VGroup(self.omega[2:30:3]).copy()
self.al_subset_fin = VGroup(self.omega[:len(self.subset_fin)]).copy()
self.shifted_subset_fin = self.subset_fin.copy()
VGroup(self.al_subset_fin, self.shifted_subset_fin).shift(shift * DOWN)
self.brace_fin = BraceText(self.al_subset_fin, "konečná")
# infinite subset
self.subset_inf = LimitSubOrdinal(
[bar for i, bar in enumerate(self.omega) if is_prime(i)]).copy()
self.al_subset_inf = self.omega.copy()
self.shifted_subset_inf = self.subset_inf.copy()
self.subset_inf.shift(shift * UP)
VGroup(self.al_subset_inf, self.shifted_subset_inf).shift(shift * DOWN)
self.brace_inf = BraceText(self.al_subset_inf, "stejná")
def construct(self):
shift = self.shift
# finite subset
self.play(
VGroup(self.omega, self.omega_brace).shift, shift * UP,
Transform(self.subset_fin, self.shifted_subset_fin))
self.wait_to(2.5)
self.play(Transform(self.subset_fin, self.al_subset_fin))
self.wait_to(7.5)
self.play(self.brace_fin.creation_anim())
self.wait_to(12)
self.play(FadeOut(VGroup(self.subset_fin, self.brace_fin))) # 13
# infinite subset
self.play(Transform(self.subset_inf, self.shifted_subset_inf)) # 14
self.play(Transform(self.subset_inf, self.al_subset_inf)) # 15
self.play(self.brace_inf.creation_anim()) # 16
self.wait_to(40.5)
self.play(FadeOut(VGroup(self.subset_inf, self.brace_inf)))
omega1 = Omega1()
omega1.shift(shift * UP)
self.play(
FadeIn(omega1),
FadeOut(self.omega),
self.omega_brace.change_brace_desc,
omega1,
"\\omega_1",
)
def construct(self):
#self.force_skipping()
question = TextMobject("What is Infinity?")
question.scale(1.3)
question.to_edge(UP)
question1 = question.copy()
self.play(Write(question))
self.dither()
infty_symbol = TexMobject('\\infty')
infty_symbol.scale(5)
self.play(Write(infty_symbol))
self.dither()
self.play(FadeOut(infty_symbol))
question2 = TextMobject("What is infinite?")
question2.scale(1.3)
question2.to_edge(UP)
self.play(ReplacementTransform(question, question2))
self.dither()
answer = TextMobject("That what is not","finite").scale(1.3)
answer.next_to(question2, DOWN, buff = 0.5)
answer_ori = answer.copy()
answer_ori.shift(DOWN)
answer_ori.highlight(BLACK)
self.play(ReplacementTransform(answer_ori, answer))
self.dither()
finite_def = BraceText(answer[-1], "possible to express\\\\ by a natural number")
self.play(finite_def.creation_anim())
self.dither(2)
dots5 = VGroup(*[Dot(color = BLUE) for _ in range(5)])
dots5.arrange_submobjects(buff = 0.5)
set5 = VGroup(VGroup(dots5), SurroundingRectangle(dots5, color = WHITE, buff = 0.3))
dots100 = VGroup(*[
VGroup(*[
Dot(color = BLUE)
for _ in range(10)
]).arrange_submobjects()
for _ in range(10)
]).arrange_submobjects(DOWN)
set100 = VGroup(dots100, SurroundingRectangle(dots100, color = WHITE))
set100.to_corner(DOWN+LEFT)
set5.move_to(set100)
brace5 = BraceText(set5, "5", UP)
self.play(FadeIn(set5), brace5.creation_anim())
self.dither()
self.play(brace5.change_desc, "finite")
self.dither()
self.play(FadeOut(brace5))
brace100 = BraceText(set100, "100", RIGHT)
self.play(
ReplacementTransform(set5, set100),
brace100.creation_anim(),
)
self.dither()
self.play(brace100.change_desc, "finite")
self.dither()
self.play(FadeOut(VGroup(brace100, set100)))
numbers = VGroup(*[
TexMobject(str(i))
for i in range(100)
])
pseudo_num = SurroundingRectangle(numbers[-1], buff = 0, stroke_width = 0, fill_opacity = 1)
numbers.gradient_highlight(BLUE, BLACK)
numbers.arrange_submobjects(buff = 0.5)
numbers.shift(-numbers.get_edge_center(LEFT))
def perspective_shift(point):
camera_distance = 10.0
ratio = camera_distance / (camera_distance + point[0])
return point*ratio
numbers.apply_function(perspective_shift, maintain_smoothness = False)
numbers.stretch_to_fit_width(2*SPACE_WIDTH-2)
numbers.center()
numbers.shift(0.5*LEFT)
rect_inf = Rectangle(width = -2*numbers.get_edge_center(LEFT)[0] + 0.5,
height = numbers.get_height() + 0.5)
set_inf = VGroup(rect_inf, numbers)
set_inf.to_edge(DOWN)
brace_inf = BraceText(set_inf, "infinite", UP, desc_scale = 1.3)
self.play(FadeIn(set_inf), GrowFromCenter(brace_inf.brace))
self.dither()
self.play(Write(brace_inf.desc))
self.dither(2)
self.revert_to_original_skipping_status()
self.play(FadeOut(VGroup(numbers, brace_inf, rect_inf, answer, finite_def)), ReplacementTransform(question2, question1))
#set0 = Square(side_length = 1).move_to(set_inf)
#brace0 = BraceText(set0, "finite", UP)
#self.play(ReplacementTransform(rect_inf, set0))
#self.play(brace0.creation_anim())
#self.dither()
#self.play(brace0.change_desc, '0')
#self.dither(2)
#self.play(FadeOut(VGroup(set0, brace0, answer, finite_def)), ReplacementTransform(question2, question1))
self.dither()
self.play(Write(TextMobject("Not just a number...")))
self.dither(3)
def construct(self):
steps_str = [
"All sets of certain property",
"Mass operation",
"New object of that property",
]
steps = VGroup([TextMobject(step_str) for step_str in steps_str])
steps.arrange_submobjects(DOWN, buff=1.5)
#steps.to_edge(UP)
arrow1 = Arrow(steps[0], steps[1])
arrow2 = Arrow(steps[1], steps[2])
self.play(FadeIn(steps[0], submobject_mode="lagged_start"))
ordinals = OrdinalClass().to_edge(DOWN)
#self.wait_to(8)
self.play(FadeIn(ordinals))
brace = BraceText(ordinals, ["Types of well-ordered", "sets"], UP)
brace.desc[-1].highlight(YELLOW)
self.play(brace.creation_anim())
#self.wait_to(22)
self.dither()
self.play(FadeOut(VGroup(ordinals, brace)))
self.play(
ShowCreation(arrow1),
FadeIn(steps[1], submobject_mode="lagged_start"),
)
all_sets1 = AllSetsPicture().scale(0.45)
all_sets2 = all_sets1.copy()
all_sets1.to_corner(DOWN + LEFT)
all_sets2.to_corner(DOWN + RIGHT)
rect1 = SurroundingRectangle(all_sets1, buff=0)
rect2 = SurroundingRectangle(all_sets2)
#self.wait_to(27)
self.play(FadeIn(all_sets1), FadeIn(all_sets2))
rectangles = all_sets1[0]
self.play(
FadeOut(VGroup(rectangles[4:])),
ReplacementTransform(VGroup(rectangles[:4]), VGroup(rect1)),
)
self.remove(rectangles)
self.play(ShowCreation(rect2))
#self.wait_to(31)
self.dither()
self.play(FadeOut(VGroup(all_sets1[1], all_sets2, rect1, rect2)))
self.play(
ShowCreation(arrow2),
FadeIn(steps[2], submobject_mode="lagged_start"),
)
p0 = steps[2].get_corner(UP + RIGHT) + 0.1 * UP
p1 = steps[0].get_corner(DOWN + RIGHT) + 0.1 * DOWN
p1 = p1 * Y_MASK + p0 * X_MASK
center = (p0 + p1) / 2 + 7 * LEFT
radius = np.linalg.norm(p0 - center)
a0 = angle_of_vector(p0 - center)
a1 = angle_of_vector(p1 - center)
angle = (a1 - a0)
arrow3 = Arc(angle, radius=radius, start_angle=a0).shift(center)
arrow3.add_tip()
arrow3.highlight(RED)
contr_label = TextMobject("Conflict!").next_to(arrow3)
#self.wait_to(37)
self.play(ShowCreation(arrow3), FadeIn(contr_label))
#self.wait_to(52)
self.dither(2)
return
self.play(FadeOut(VGroup(
arrow1,
arrow2,
arrow3,
contr_label,
steps,
)))