def show_natural_loggedness(self, expression):
base, constant = expression[1], expression[-3]
log_constant, exp_constant = constant.copy(), constant.copy()
log_base, exp_base = base.copy(), base.copy()
log_equals, exp_equals = map(TexMobject, "==")
ln = TexMobject("\\ln(2)")
log_base.move_to(ln[-2])
ln.remove(ln[-2])
log_equals.next_to(ln, LEFT)
log_constant.next_to(log_equals, LEFT)
log_expression = VGroup(
ln, log_constant, log_equals, log_base
)
e = TexMobject("e")
exp_constant.scale(0.7)
exp_constant.next_to(e, UP+RIGHT, buff = 0)
exp_base.next_to(exp_equals, RIGHT)
VGroup(exp_base, exp_equals).next_to(
VGroup(e, exp_constant), RIGHT,
aligned_edge = DOWN
)
exp_expression = VGroup(
e, exp_constant, exp_equals, exp_base
)
for group, vect in (log_expression, UP), (exp_expression, DOWN):
group.to_edge(RIGHT)
group.shift(vect)
self.play(
ReplacementTransform(base.copy(), log_base),
ReplacementTransform(constant.copy(), log_constant),
run_time = 2
)
self.play(Write(ln), Write(log_equals))
self.dither()
self.play(
ReplacementTransform(
log_expression.copy(),
exp_expression,
run_time = 2,
)
)
self.dither(2)
ln_a = expression[-1]
self.play(
FadeOut(expression[-2]),
FadeOut(constant),
ln_a.move_to, constant, LEFT,
ln_a.highlight, BLUE
)
self.dither()
self.play(*map(FadeOut, [log_expression, exp_expression]))
self.dither()
def show_natural_loggedness(self, expression):
base, constant = expression[1], expression[-3]
log_constant, exp_constant = constant.copy(), constant.copy()
log_base, exp_base = base.copy(), base.copy()
log_equals, exp_equals = map(TexMobject, "==")
ln = TexMobject("\\ln(2)")
log_base.move_to(ln[-2])
ln.remove(ln[-2])
log_equals.next_to(ln, LEFT)
log_constant.next_to(log_equals, LEFT)
log_expression = VGroup(ln, log_constant, log_equals, log_base)
e = TexMobject("e")
exp_constant.scale(0.7)
exp_constant.next_to(e, UP + RIGHT, buff=0)
exp_base.next_to(exp_equals, RIGHT)
VGroup(exp_base, exp_equals).next_to(VGroup(e, exp_constant),
RIGHT,
aligned_edge=DOWN)
exp_expression = VGroup(e, exp_constant, exp_equals, exp_base)
for group, vect in (log_expression, UP), (exp_expression, DOWN):
group.to_edge(RIGHT)
group.shift(vect)
self.play(ReplacementTransform(base.copy(), log_base),
ReplacementTransform(constant.copy(), log_constant),
run_time=2)
self.play(Write(ln), Write(log_equals))
self.dither()
self.play(
ReplacementTransform(
log_expression.copy(),
exp_expression,
run_time=2,
))
self.dither(2)
ln_a = expression[-1]
self.play(FadeOut(expression[-2]), FadeOut(constant), ln_a.move_to,
constant, LEFT, ln_a.highlight, BLUE)
self.dither()
self.play(*map(FadeOut, [log_expression, exp_expression]))
self.dither()
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)
def construct(self):
#self.force_skipping()
bubble = SVGMobject(file_name = "thought_bubble",
initial_scale_factor = 0.31)
bubble.set_color(WHITE)
bubble.stretch(1.25, 0)
bubble.to_corner(DOWN+RIGHT)
self.play(ShowCreation(VGroup(*bubble[:2])))
self.play(ShowCreation(bubble[2]))
self.dither()
subbubbles = self.make_fractal(bubble[-1], [0.8*LEFT, 0.8*RIGHT])
self.play(ShowCreation(subbubbles, submobject_mode = "all_at_once", run_time = 2))
self.dither()
main_rect = Rectangle(width = 2*SPACE_WIDTH-1, height = 2*SPACE_HEIGHT-1)
rect_template = main_rect.copy()
rect_template.stretch_about_point(0.5, 1, rect_template.get_edge_center(UP))
subrects = self.make_fractal(rect_template, [0.9*LEFT, 0.9*RIGHT], ratio = (0.45, 0.8, 0.6))
self.play(ReplacementTransform(bubble[-1], main_rect),
ReplacementTransform(subbubbles, subrects),
bubble[0].behind_edge, DOWN,
bubble[1].behind_edge, DOWN,
run_time = 2)
#self.dither()
implications = TexMobject("\Rightarrow\Leftarrow")
implications.arrange_submobjects(DOWN, buff = 0.5)
implications.rotate(-np.pi/2)
contradiction = VGroup(
TexMobject("X\\in X"),
implications,
TexMobject("X\\not\\in X"),
)
contradiction.arrange_submobjects(DOWN)
contradiction.to_corner(LEFT+DOWN, buff = 1)
contradiction.shift(RIGHT)
contradiction.add(implications[1])
implications.remove(implications[1])
self.revert_to_original_skipping_status()
#self.dither()
self.play(FadeIn(contradiction, submobject_mode = "lagged_start"))
#self.dither()
questionmarks = TextMobject("???")
questionmarks.next_to(contradiction, buff = 0.5)
#self.play(Write(questionmarks))
#self.dither()
#contradiction.add(questionmarks)
cross = Cross().set_color(RED)
cross.scale((
contradiction.get_width() / cross.get_width(),
contradiction.get_height() / cross.get_height(),
1,
))
cross.move_to(contradiction)
title = TextMobject("Set Theory").scale(1.3).to_edge(UP)
division = DashedLine(title.get_edge_center(DOWN) + 0.2*DOWN, SPACE_HEIGHT*DOWN)
self.play(
contradiction.highlight, DARK_GREY,
ShowCreation(cross),
)
axioms = TextMobject("Axioms")
axioms.next_to(division)
self.play(Write(axioms))
self.dither()
left_rect, right_rect = [], []
for rect in subrects:
if rect.get_center()[0] < 0: left_rect.append(rect)
else: right_rect.append(rect)
left_rect = VGroup(*left_rect)
right_rect = VGroup(*right_rect)
self.play(FadeOut(VGroup(right_rect, main_rect, contradiction, cross)))
self.play(
ShowCreation(division),
UnapplyMethod(title.behind_edge, UP),
left_rect.to_edge, DOWN, 0.8,
)
self.dither()
theory_of_infinity = TextMobject("Theory of Infinity").highlight(BLUE)
foundations_of_mathematics = TextMobject("Foundations of\\\\ Mathematics",
alignment = "\\raggedright")
theory_of_infinity.next_to(division, LEFT, aligned_edge = UP)
foundations_of_mathematics.next_to(division, RIGHT, aligned_edge = UP)
VGroup(theory_of_infinity, foundations_of_mathematics).shift(0.2*DOWN)
self.play(FadeIn(theory_of_infinity), run_time = 2, submobject_mode = "lagged_start")
self.dither()
self.play(FadeIn(foundations_of_mathematics), run_time = 2, submobject_mode = "lagged_start")
self.dither()
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))
